I managed to do something. What does it mean?
I've heard that "managed to" may has another meaning besides like to control anything used in the past tense.
Has that second meaning appeared like analog of "learned to" and it uses in informal talkings?

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It means that you did something that was somewhat difficult.
For example - 
"I managed to finish my homework."  means "I did my homework and it was difficult."  (likely it was hard, or there was very little time, or there was a lot of homework)
"I managed to climb the mountain" means "I climbed the mountain and it was difficult to climb."

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"I managed to" = "I was able to"
It does often imply that whatever was done was difficult or complicated.