What is a motion to dismiss?

Answer 1

A motion is a request to the court to do (or not do) something.

Dismiss means to throw out.

With prejudice means it can not be brought again.

If dismissed without prejudice, it can be brought again at a later time.

So, a motion to dismiss with prejudice is a request to the court to throw out the issue and never let it be brought again.

Answer 2

It's a writ, usually filed by a trial counsel, asking the court to dismiss a charge or complaint against a person due to factors named in the writ. Like the lack of evidence, or failure of the complainant to appear as an example.
A Motion for Dismissal is drawn up after a Complaint is received when the Defendant has reason to believe the Complaint is invalid.

There are certain reasons that courts will accept such as the wrong venue; the Complaint is against someone who is not a party to it; if the Complaint was not served properly.

It is usually best not to ask for a dismissal if the cause is not one of the above because it shows your hand. The other side can just do a revised complaint and you're right back where you started. You could better just do an Answer to the Complaint and have one of your answers reflect the reason you think the case is not valid.