(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 ___" or "were ___":

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":

Is that guy waving to you?

You can also "wave at" someone. "Waving at" someone has a little bit broader meaning. It can mean waving to somoene 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?

(someone) seemed to (do something)

When you want to guess about other people's emotions or intentions in the past, you say that they "seemed to" do something:

Everyone seemed to have a good time at Ella's party, wouldn't you say?

You seemed to be confused earlier when I was giving the instructions. Do you have any questions?

Will seemed to like the gift you gave him.

Tne negative version of "seemed to" can be either "didn't seem to ___" or "seemed not to ___":

They didn't seem to hear what I said.

They seemed not to hear what I said.

"Seemed not to ___" is a little more formal.