Difference between a request, an offer, and a suggestion

A REQUEST  – a polite or formal demand for something.

They have made an urgent request for international aid.
Anderson repeated his request that we postpone the meeting.
The study was done at the request of the Chairman.
Further details will be sent on request.
There were no flowers at the funeral, by request.

AN OFFER  – to ask someone if they would like to have something, or to hold something out to them so that they can take itoffer somebody something.

Can I offer you something to drink?
They offered him a very good job, but he turned it down.  
The drama school offers places to students who can show talent.
I don’t need any help, but it was nice of you to offer.
My dad has offered to pick us up.
The newspaper offered to apologise for the article.
Your doctor should be able to offer advice on diet.
The course offers the opportunity to specialize in the final year.
A number of groups offer their services free of charge. The Centre offers a wide range of sports facilities.
I did what I could to offer comfort to the family.

A SUGGESTION  –  an idea, plan, or possibility that someone mentions, or the act of mentioning it.

Any suggestions would be welcome.
The committee rejected the suggestion that houses should be built on this site.
Here are some suggestions for further activities.
My suggestion about menus was ignored.
She took a seat at his suggestion.