The basic meaning of the word "encounter" is "meet". But you usually don't use "encounter ___" to talk about meeting people. Instead, you "encounter" things like problems, words, and situations:

The study indicates that foreign students encounter a variety of challenges in their new environment, including language difficulties, discrimination, and bullying.

If I ever encountered a situation where I had to choose between my marriage and my career, I'm not exactly sure what I would do.

In the situation above, the speaker could also say:

I've never come across that phrase before.

"Come across ___" is a more casual way to express the idea of "encountering" something like a word or phrase. To "come across" something means to encounter it or to find it by accident:

Sometimes you'll be flipping through the channels and come across something really interesting.