A paid product review involves you (or a member of your team) testing out a product/service and then writing/recording a full review about it. Payment for this could come in various forms, including cash or a free product.
Advantages: This works really well if your opinion within your niche is trusted. I do quite a lot of these types of posts for SEO/social media tools; however, I don't accept money for them. Instead I ask that I have free access to their software or a free copy for my awesome readers!
Getting free stuff is always great but if you're also able to run a giveaway on your blog with a prize that someone else has provided then that's even better.
Disadvantages: I've always found this method a bit of a tricky thing to balance from an impartiality point of view. If I'm going to write a review (paid or not) then I'm always going to be completely honest, and I make this clear to the company beforehand.
Now, companies don't really like you bad-mouthing their products at their expense - so you have to really walk the tightrope here! Striking the balance of giving the company ROI and delivering honest and useful information to your readers is very tricky, so make sure you plan this out well.