The following are considered assignments:
When you do a product search in the review category under the Account tab, you are given three options: Manufacturer, Eden, and Buy-out.
Buy-out assignments can be used on any item, anywhere on the site so long as it is not currently on sale and you will be given a 30% discount. Simply go to the product page and click "Grab It" and then submit the request through the assignment cart. If you're in the product search under the Account tab, click Submit request and it will tell you to go to your assignment cart. It doesn't matter how many reviews have been completed by other users, you can still get the item for review at a discount.
Manufacturer sponsored reviews can be found under the product search in the Account tab and those are sponsored directly by the manufacturer. You can request as many of these per month as you'd like, providing you do not already have an open assignment. Look for items with less than three reviews.
Eden sponsored reviews are a once a month deal and are a part of the advanced reviewer status. These can also be found on the Account/Review/Product search. Just make sure you pick something with less than three reviews.
Any item you purchase outright using points/gift cards or your own money can also be reviewed and these do not count toward open assignments. It's up to you whether or not you choose to review something you bought just because you wanted it.
You have three weeks to complete each assignment.
NOTE: Lingerie CANNOT be requested as a sponsored assignment. Lingerie can be requested as a buy-out assignment or you can purchase an item and review it at your leisure.
I hope this clears things up!