It´s used to express the meanings of permission, guessing and possibility. In Turkish, you can conjugate this auxilarity verb with every tense. 

Let´s show affirmative and negative types with present simple tense.

Negative type of "-ebilmek" is "-ememek". Its own is in present simple tense directly.

-eme+yor-----> -emiyor

Do you remember the fact that "-y" of "yor" is making the large vowel narrow?

If you want, let´s show a few tense more and then let´s start to give example sentences.

And let´s also give an example with compound tenses. I´m going to prefer to write the extra verb as suffix, i´m not going to write seperately.

etc. etc. etc. ....

Like you have seen until now, you should think the verb "-ebilmek, -ememek" exactly a normal verb. Because -like you see- they can be conjugated with all the tenses.

If you say "-ebildim, -emedim", you mean "I succeeded / didn´t succeed to ...". 

If you say "-ebilirdim, -emezdim", you mean "I had/hadn´t possibility/opportunity to ...". Also, it can be used as ability meaning of "used to".

yardım etmek (to help)

valiz (the suitcase)

kesinlikle (absolutely, certainly)

taşımak (to carry)

yardım et+me+z ise+n----------------->yardım etmezsen (if you don´t help)

bu valiz (this suitcase)

bu valiz+i(acc.)--------->bu valizi (this suitcase)

taşı+ama+m------------>taşıyamam (i can´t carry)

Yardım etmezsen bu valizi kesinlikle taşıyamam (I can never carry this suitcase if you don´t help.)

....................the meaning of ability

sır (the secret)

vermek (to give)

sır vermek (to confide, to tell a secret)

-den başka, ... hariç (except ...)

hiç kimse (nobody)

sana (to you)

ver+dik+m sır------------------------->verdiğim sır (the secret I told/tell)

the original type of "sır" is "sırr". therefore the "r" which had disappeared will return when a vowel-started suffix.

sır+ı(accusative)---------------------->sırrı (the secret)

sana verdiğim sırrı ((accusative) the secret I told to you)

sen+den başka----------------------->senden başka (except you)

senden başka hiç kimse (nobody except you)

hiç kimse+e(dative)------------------->hiç kimseye (to nobody)

ver+eme+z idi+m---------------------->veremezdim (i couldn´t give)

Sana verdiğim sırrı senden başka hiç kimseye veremezdim. (I couldn´t have told the secret I told to you to nobody except you)

.....................................the meaning of ability

niye, neden (why)

otel (the hotel)

kalmak (1to stay, 2to remain, 3..4...)

otel+de---------------------------->otelde (at the hotel)

kal+dı+nız-------------------------->kaldınız (you stayed)

biz+de----------------------------->bizde (at us)

biz+de de-------------------------->bizde de (at us too, also at us)

kal+abil+r idi+niz-------------------->kalabilirdiniz (you could have stayed)

Neden otelde kaldınız? Bizde de kalabilirdiniz. (Why did you stay at a hotel? You could have stayed at us too.)

başlangıç (the beginning)

seviye, düzey (the level, the grade)

orta (1middle, 2intermediate, 3..4..)

yaklaşmak (to draw near,to draw close)

hatta (what´s more, moreover) (pronunciation: hat´taa)

ulaşmak (to arrive, to reach)

-mek üzere (about to ... , be going to ... soon)

artık (1from now on (direct: after now), 2anymore, 3the waste)

uzun (1long, 2tall)

cümle (the sentence)

kurmak (1to set, to set up, 2to found, 3to form, 4to assemble)

cümle kurmak (to make a sentence)

başlangıç seviye+i(poss.)---------->başlangıç seviyesi (the beginner level (direct: the beginning level))

başlangıç seviye+i(poss.)+de(loc.)---------->başlangıç seviyesinde (in the beginner level)

başlangıç seviye+i(poss.)+de+idi(past)+m--->başlangıç seviyesindeydim (i was in the beginner level)

orta seviye (the intermediate level)

orta seviye+e(dative)---------------------->orta seviyeye (to the intermediate level)

yaklaş+yor+im----------------------------->yaklaşıyorum (i´m drawing nearer)

ulaşmak üzere+im-------------------------->ulaşmak üzereyim (i´m about to reach)

orta seviyeye ulaşmak (to arrive at the intermediate level)

orta seviyeye ulaşmak üzereyim (i´m about to/going to arrive soon)

daha uzun (longer)

cümle+ler------------------------->cümleler (the sentences)

daha uzun cümeler (longer sentences)

kur+abil+ecek+im------------------>kurabileceğim (i will be able to set)

Başlangıç seviyesindeydim ama orta seviyeye yaklaşıyorum, hatta ulaşmak üzereyim. Artık daha uzun cümleler kurabileceğim. (I was in the beginner level but I´m drawing nearer to the intermediate level, moreover, I´m about to arrive. I will be able to make longer sentences from now on.)

...............the meaning of ability

iyi (good, well)

duymak (to hear)

olmak (1to be, 2....3....11..14..)

yer (1the place, 2the floor, the ground, 3the earth, 4the position, 5..6..)

gürültü (the noise)

geçmek (to pass)

sen+i(acc.)------------------------------>seni (you (accusative))

duy+ama+yor+im------------------------->duyamıyorum (i can´t hear (present continuous))

iyi duyamıyorum (i can´t hear well)

ol+dik+n yer----------------------------->olduğun yer (the place where you are)

gürültü+li-------------------------------->gürültülü  (noisy)

gürültü+siz------------------------------>gürültüsüz (noiseless)

bir yer (somewhere, whichever place)

gürültüsüz bir yer (a noiseless place)

gürültüsüz bir yer+e geç+- --------------->gürültüsüz bir yere geç (go noiseless somewhere (direct: pass to a noiseless place))

Seni iyi duyamıyorum. Olduğun yer çok gürültülü. Gürültüsüz bir yere geç. (I can´t hear you well. The place where you are in is too noisy. Go a noiseless somewhere.)

...............the meaning of ability

saat (1the clock, o´clock, 2the hour, 3the watch)

iki (two)

-e kadar (till, until)

dışarı (1outside, 2outdoor)

çıkmak (1to go out, 2to ascend, 3...4....7to flirt, 8to be understood suddenly,9...)

iki+e kadar------------------------------->ikiye kadar (until two)

çık+ama+zsınız--------------------------->çıkamazsınız (you can´t go out)

Saat ikiye kadar dışarı çıkamazsınız. (You aren´t allowed to/can´t go outside until two o´clock.)

=It would be the same if we added dative case suffix to "dışarı".

(dışarı+a-->dışarıya (direct: to the outside))

...........meaning of permission

bugün (today)

hava (1the weather,.......)

yağmur (the rain)

olmak (1to be, ........)

yağmur+lı-------------------------------->yağmurlu (rainy)

ol+abil+r--------------------------------->olabilir (it can/may be)

Bugün hava yağmurlu olabilir. (The weather may be rainy today.)

...........meaning of guessing/possibility.

How to make interrogative?

The answer isn´t difficult. These tenses we are conjugating "-ebilmek" and "-ememek" with are ultimately the tenses you had already learnt. If you don´t remember how to make interrogative them, you can look at their lessons. However it requires to write a few example for interrogative sentences too. 

bura (this place, here)

oturmak (1to sit, 2to sit down, 3to dwell, to live, 4to be suitable, 5...6...)

bura+a(dative)--------------------------->buraya (to this place, here)

otur+abil+r mi+iz------------------------->oturabilir miyiz? (can/may we sit/sit down?)

Buraya oturabilir miyiz? (May we sit down here?)

........meaning of permission

yirmi (twenty)

yaş (1wet, 2the age (the period of the life for each one year))

konser (the concert)

yalnız (1alone, lonely ,2lone, 3but)

gitmek (to go)

yaş+ı+da+ım----------------------------->yaşındayım (i´m ... years old)

konser+e-------------------------------->konsere (to the concert)

git+eme+ecek mi+im--------------------->gidemeyecek miyim? (won´t i be able to go?)

Yirmi yaşındayım. Konsere yalnız gidemeyecek miyim? (I´m twenty years old. Won´t I able to go to the concert alone?)

......meaning of ability

geçen (last, previous (direct: the one that has/had passed)

yıl, sene (the year)

turist (the tourist)

gelmek (to come)

geçen yıl (last year)

çok turist (many tourists)

gel+miş+di------------------------------->gelmişti (he had come)

bu yıl (this year)

gel+me+mek----------------------------->gelmemek (not to come)

gel+me+ebil+r----->gelme+ebil+r--------->gelmeyebilir (perhaps he doesn´t come)

Geçen yıl çok turist gelmişti ama bu yıl gelmeyebilir. (Many tourist have come last year but perhaps they don´t come this year.)

..........meaning of possibility

mutlaka (certainly)

yemek (1to eat, 2the meal)

istemek (to want)

yemek zorunda değilsin (you don´t have to eat)

iste+me+yor ise+n--->iste+me+yor+se+n--------->istemiyorsan (if you don´t want)

ye+me+ebil+r+sin------------------------------->yemeyebilirsin (it can be supposed this way: you have possibility/right not to eat. : you are allowed not to eat.)

Mutlaka yemek zorunda değilsin. İstemiyorsan yemeyebilirsin. (You don´t have to eat certainly. You are allowed not to eat if you don´t want.)

........meanings of possibility and permission are together. 

bura (this place, here)

olmak (to be...)

söylemek (to say)

inanmak (to believe)

bura+da(locative)------------------------>burada (here)

ol+dık+m+ı(acc.)------------------------->olduğumu ((acc.)the fact that i am/was)

söyle+me+- ----------------------------->söyleme (don´ t say)

inan+ma+abil+r+ler----------------------->inanmayabilirler (perhaps they don´t believe)

Burada olduğumu söyleme, inanmayabilirler. (Don´t say I am here, perhaps they don´t believe.)

...........meaning of possibility/guessing.

elektrik (the electric)

kesilmek (1to be cut, 2to go off (electric), 3..4..5.)

rapor (the report)

bu gece (tonight)

yazmak (to write)

söz vermek (to promise)

kesil+me+z ise-------->kesilmez ise------>kesilmezse (if it´s not cut)

rapor+u(accusative)-------------------->raporu (the report (acc.))

yaz+r+ım------------------------------->yazarım (i write)

kesil+ebil+r----------------------------->kesilebilir (it may be cut)

kesilebilir de (it aslo can/may be cut)

söz ver+eme+m------------------------->söz veremem (i can´t promise)

Elektrik kesilmezse raporu bu gece yazarım ama kesilebilir de, söz veremem. (I write the report tonight if the electric doesn´t go off but it also may go, I can´t promise.)

..............meaning of possibility

alıştırma (the exercise, the practice)

son (last)

beş (five)

paragraf (the paragraph)

anlamak (to understand)

bilmek (to know)

ki (that)

ileri (1forward, 2advanced)

seviye, düzey (the level)

ulaşmak, varmak (to arrive, to reach)

alıştırma+da+ki---------------------->alıştırmadaki (that/which is in the exercise)

son beş paragraf (last five paragraphs)

alıştırmadaki son beş paragraf (the last five paragraphs in the exercise)

alıştırmadaki son beş paragraf+ı(ac.)->alıştırmadaki son beş paragrafı

anla+abil+yor ise+n--->anlayabiliyor isen-->anlayabiliyor+se+n->anlayabiliyorsan (if you can understand (present continuous))

bil ki (know that ..)

ileri seviye+e------------------------>ileri seviyeye (to the advanced level)

ulaş+mış+sın------------------------>ulaşmışsın (you have arrived)

Alıştırmadaki son beş paragrafı anlayabiliyorsan bil ki ileri seviyeye ulaşmışsın. (Know that you have arrived at the advanced level if you can understand the last five paragraphs in the exercise.)

..............meaning of ability