Ability means the possibility an agent has to perform an action, due to factors internal to him/her.
This ability may be more or less circumstantial, seen mainly as relevant to the current situation, or conversely, persistent, that is seen as having duration beyond the current moment, mainly because having been acquired through a process of learning, whether formal or informal.

● The construction "verb-A OL-", i.e. the present gerund followed by the auxiliary verb ol- (root meaning: 'to take'), is the main form used to express ability, and is the equivalent of 'can' or 'be able to'.
It expresses mainly circumstantial ability, but can be used for permanent or acquired ability as well.

- Circumstancial ability:

Xatni bugun yozolmadim.
I could not write the letter today.

Bugun ishga bora olmadim.
Today, I couldn't go to work.

O'qiy olmadi.
He was not able to read.

U mening oldimga kela oldi.
He was able to come to me.

- Permanent ability:

Men bu mashinamni tuzata olaman.
I can repair this car of mine.

Biz haqiqiy mustaqilikni demokratiyasiz tasavvur qila olmaymiz.
We can't imagine a real independence without democracy.

Ko'ra olmas edilar.
They (habitually) weren't able to see.

Men qilola turibman.
I am continuing to be able to do.

- Acquired ability, with about the same meaning as in the constructions with bil-, 'to know' (see below):

Har bir o'zbek qizi ovqatlarni pishira oladi.
Every Uzbek girls are able to cook meals.

Biz o'zbekcha kitoblarni o'qiy olamiz.
We are able to read books in Uzbek.

- Performative modality: This construction with ol- may express (in the negative form) a polite refusal, or a polite request, asking for permission, when used in the interrogative form:

Men buni butunlay qilolmayman.
I might not be able to do it thoroughly.

Nodirani chaqira olasizmi?
Could you call Nodira?

Siz mashinamni tuzata olasizmi?
Could you repair my car?

● The constructions "verb-ISHNI BIL-", i.e. the infinitive in the accusative case, followed by the verb bil- (root meaning: 'to know'), and "verb-A BIL-", i.e. the present geround followed by the auxiliary bil-, are the main form used to express acquired ability, acquired mainly through learning, and are the equivalent of 'know how to' (or 'can' with this meaning).
The first of these constructions might be used slightly more than the second one.

- The form with the infinitive:

O'g'ling o'qishni biladimi?
Your son can read?

Suzishni bilasizmi?
Can you swim?

Bu eshitishni biladigan odam.
He is somebody who does not know how to listen.

- This form can have also the more direct meaning of 'to know how to' without necessarily involving a knowledge acquired through learning:

Qanday borishni bilasizmi?
Do you know how to get there?

- The form with the present gerund:

Chap qo'lim bilan yozabilaman.
I can write with my left hand.

Yoza bilmayman.
I do not know how to write.

Qila bilasa kerak edi.
He may have been able to do.

● The nominal predicate based on qodir ('capable') expresses the inner capacity to do something.
It is used with the infinitive form of the verb in the dative case.

Bu sportchi qisqa masofaga yugurishda yuqori ko'rsatkichlarga erishishga qodir.
This sportsman is capable of high successes in spring.

U bu ishni bir o'zi bajarishga qodir.
She is capable to fulfil this work alone.

U zafar qozonishga qodir.
He is capable to win.

- Other nominal predicates with the same meaning are qobiliyat and layoqat.

U kishida bu ishni qilish uchun qobiliyat bor.
That man has the ability to do this work.

U matematikaga qobiliyati borligini ko'rsatdi.
He shows aptitude for mathematics.

Unda chet tillarga layoqat bor.
He has gift for foreign languages.

Bu ishni qilish uchun unda layoqat bor.
He has the ability to do this work.