These words have similar meanings, but they are used a little differently.

Usage as verbs 

“Request” means “ask for”, NOT “ask”.

You “ask a question”, but you can’t “request a question”. E.g.: 

WRONG
He requested whether I’d read the email.
I have requested Mr Johnson what he thinks of the idea.
They requested how he will carry out the plan.

RIGHT
He asked whether I’d read the email.
I have asked Mr Johnson what he thinks of the idea.
They asked how he will carry out the plan.

You “request something”, and you “ask for something”. E.g.:

He asked for more time to read the contract.
He requested more time to read the contract. 

The authority asked for a response by 26 May.
The authority requested a response by 26 May.

But you cannot “request for something”. E.g.: 

WRONG
In your letter of 5 February you requested for additional documents concerning the claims. 

RIGHT
In your letter of 5 February you asked for additional documents concerning the claims.
In your letter of 5 February you requested additional documents concerning the claims.

Similarly, you “ask somebody for something”, but you “request something from somebody”. E.g.:

The client asked us for a 2-page summary of the report.
The client requested a 2-page summary of the report from us. Or …requested us to provide… 

Where is the 2-page summary of the report that the client asked us for?
Where is the 2-page summary of the report that the client requested from us?

However, both verbs follow the same pattern if you “ask somebody to do something” or “request somebody to do something”:

Mr Smith asked me to post the letter when I left the office.
Mr Smith requested me to post the letter when I left the office.

Both verbs also follow the same pattern with “that”:

The client asked that his complaint be officially registered with the authorities.
The client requested that his complaint be officially registered with the authorities.

“Ask” is everyday English, “request” is formal.

Usage as nouns

“Request” can also be used as a noun:

The meeting was postponed at the client’s request.
The court refused our request for an additional witness at the hearing.

It is not usual to use “ask” as a noun. But you may hear it used as a noun in informal speech. E.g.:

So what’s the ask? (i.e. What are they asking us to do?)

The word is commonly collocated with “big” or “tough”, e.g.: 

I’ve got a really big ask for you. Could you come into the office and help me out on Sunday?

It’s a tough ask, but if we all work hard I believe we can do it.

Remember that this should only be used in informal speech. Do not use it in writing.