What is the difference between suggested, client and assignment contacts?

You may see three different sections on an assignment contact screen.

Suggested contacts are people that have worked at the company at some point but are not actually listed as being part of the company, probably because the company was added to the system after the person profile.  If you tick the checkbox on a suggested contact, you will get the option to add it to the company, making them a company contact.

Client contacts are the same as company contacts - clients are companies that have an assignment on the system, so (as you are looking at an assignment page) they are called client contacts.  These are people you know of that work at the company and are related to the company record.   In the assignment/client contact section, if you select the checkbox on a client contact, you can relate that person as an assignment contact. 

Assignment contacts are a subset of client contacts - these are people at the company that are specifically involved with that assignment.