“We got tons of responses to that ad I posted.”

You're making a short film. You needed an actor, so you placed an ad calling for actors on a website. A lot of people responded. You say this when telling your assistant director about it.

We got tons of responses to that ad I posted.

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(someone) got responses to (something)

You send or post a message to people. Then people respond to you (meaning they send a message back to you). You get responses from them.

If you send out the invitations now, you'll probably start to get responses by the beginning of August.

You identify your original message with "to":

I haven't gotten any responses to my email yet.

tons of (something)

"A ton" is literally a weight that's equal to 2,000 pounds (910 kilograms).

In normal conversation, when you talk about "tons of" something, it's usually an exaggeration that really means "a lot of".

Don't worry, we've got tons of time before takeoff.

I ate tons of food.

This is a casual way of speaking, and is usually used by younger people. It can be used in blog posts and when chatting with friends, but not in formal writing for school.

that (thing)(someone did)

In the example above, the listener knows about the ad. But she might get confused if the speaker just said:

We got tons of responses to the ad.

So the speaker says "that ad I posted" instead. You use this phrase to talk about things that the listener remembers but needs a little extra information about:

Do you remember that song we wrote together back in college?

I ran into that painter we met at Janice's house today.

post an ad

To "post an ad" means to place an advertisement on a website, in a newspaper, or on a board in a public place.

You use the word "post" with cheap ads that individual people make. When a more serious ad for a larger company has an ad, they "run" it:

Toyota has been running ads in magazines and newspapers touting their efforts to improve the safety of their vehicles.