“I waved to him but he didn't seem to recognize me.”

Something interesting happened to you yesterday while you were having lunch with your friend Amy. Now you are telling the story to another friend who wasn't there. You have explained the situation and now you're starting to talk about the interesting thing that happened. You say:

I waved to him but he didn't seem to recognize me.

(someone) (did something)

When you're telling the interesting or important events in a story, you use the simple past tense:

He said something to me.

They fell down the stairs.

We laughed so hard we couldn't stand up any more.

This is different from when you're explaining the background at the beginning of a story. When you're doing that, you use "was ___":

He was saying something to me.

We were laughing really hard.

Notice that you can't say "They were falling down the stairs" because falling is too short and too interesting to just be the background situation of a story.

wave to (someone)

To "wave to" someone means to shake your hand back and forth to say "hello".

You can also "wave at" someone, but that can be done to mean "hello", "goodbye", or other meanings as well.

recognize (someone)

To "recognize" someone means to know who they are when you see them. You can recognize a friend or an acquaintance, if they look different than usual or you didn't see them very clearly.

Joe's gotten so thin, I almost didn't recognize him.

You can also recognize a person you don't know, if they're famous or you've seen their picture somewhere.

I recognize you from somewhere. You were on American Idol, weren't you?