〜える / 〜られる

can do

Says what someone can do or what is possible: "can" or "be able to".

Readings

Conjugation rules

For godan verbs, change the last う-sound to its え-sound, then add る: 書(か)く becomes 書(か)ける.
For ichidan verbs, replace る with られる.
する becomes できる, a different verb, and 来(く)る becomes 来(こ)られる (irregular). Noun + する verbs use できる too: 勉強(べんきょう)できる.
The result is an ichidan verb: 書(か)けます, 書(か)けない, 書(か)けた.

godan

書(か)く(to write)書(か)ける

話(はな)す(to speak)話(はな)せる

ichidan

食(た)べる(to eat)食(た)べられる

irregular

する(to do)できるnot しられる

勉強(べんきょう)する(to study)勉強(べんきょう)できる

来(く)る(to come)来(こ)られる

愛(あい)する(to love)愛(あい)せるnot 愛(あい)できる

Some する verbs with a one-kanji stem, like 愛(あい)する, take the godan ending, as if the verb were 愛(あい)す.

日本語(にほんご)が話(はな)せます。

I can speak Japanese.

この漢字(かんじ)が読(よ)めますか。

Can you read this kanji?

The thing you can do is usually marked with が rather than を: 日本語(にほんご)が話(はな)せます. を is also heard, especially in longer sentences.
In casual speech ichidan verbs and 来(く)る often drop the ら: 食(た)べれる, 来(こ)れる. This is very common in conversation, but careful writing keeps られる.
見(み)える and 聞(き)こえる mean something is visible or audible on its own. 見(み)られる and 聞(き)ける mean you have the chance or ability to see or hear it.

〜ことができるcan / be able to
The same meaning with the dictionary form + ことができる; a little more formal.

〜れる / 〜られるpassive
The ichidan passive has the same shape, 食(た)べられる, but means "is eaten".
〜れる / 〜られるlight honorific
The same れる・られる can show respect for the person doing the action.