“I saw you checking out that lady in there.”
When you were at a restaurant with your dad, an attractive woman was sitting at a table nearby. You noticed that your father was looking at her a lot. After you leave the restaurant, you say this to tease him for looking at the woman.
I saw you checking out that lady in there.
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This means to see someone when they're doing something:
I just saw Judy talking to Tom in the hallway.
To "check (someone) out" means to look at a person's whole body to see how attractive they are. You can check yourself out in a mirror, or check someone else out. You can also "check out" a certain part of a person's body:
Were you checking out my ass?
This is very casual English, of course.
You can also "check out" an object, an e-mail message, a video, and so on:
Check out these shoes I bought. Nice, huh?
But this just means to look at something, and doesn't have a negative sexual meaning.
"Lady" is another word for "woman".
The word "lady" sometimes sounds more respectful than "woman". You can use it to talk about someone who you don't know:
I met a lady on the bus who said that she was a physical therapist.
It's not very polite to use the word "lady" to directly address someone. It can actually sound rude:
Get out of my way, lady!
"Miss" or "ma'am" are more polite:
Excuse me, miss. I think you dropped something.
But it's OK to call a group of women "Ladies":
Right this way, ladies. Follow me.