There are many different types of coupons. The 2 main types of coupons are Manufacturers coupons and Store Coupons.
Manufacturer Coupons are produced by the brand manufacturer and are there to encourage buyers to purchase their products. When you use a manufacturer coupon at a store you will have the face value deducted from your overall total at the checkout. The store will then send the coupon back to the manufacturer and be reimbursed for the face value of the coupon, plus a small handling fee.
A Store Coupon is produced by a specific retailer. These coupons can typically only be used at the retailer that produced them, however there are some grocery chains that accept competitors coupons. The coupon will usually have the retailers logo on it, and will say "Store Coupon" at the top versus "Manufacturers Coupon"
Store coupons and manufacturer coupons can almost always be stacked. If you have a $1.00 off Manufacturers coupon for a jar of peanut butter, and a $0.50 off store coupon for that exact same product, you can "stack" them to maximize your savings, in this case saving you $1.50.
Other types of coupons:
Catalinas: Catalinas are the coupons that print out after your receipt. Typically, they will print coupons based on your purchase history on your store card, or they could be for items they think you would be interested in purchasing. Although they say "Manufacturers Coupon" on them (and can be treated as such), they are typically referred to as catalinas because of where they come from.
Peelies: Peelies are manufacturer coupons that are stuck to the product when you pull it off the shelf. They typically say 'SAVE NOW' and can be used at checkout.
Blinkies: Blinkies are the little machines in the isles with the little blinking light that produces manufacturer coupons. I loved these as a kid for all the wrong reasons.
Printable Coupons: Printable Coupons are typically manufacturer coupons that you can print from coupons.com, smartsource.com and redplum.com - or other online coupon sites. They are treated just like manufacturer coupons, however, always check your stores coupon policy. Many folks have forged coupons, so retailers have limited the amount of printable coupons you can use, if any.
eCoupons: eCoupons are coupons that you can load onto your store loyalty cards from the retailer website. When you get to the checkout, you swipe your loyalty card and it will deduct the amount automatically from your total. Harris Teeter is a local grocery store here and they provide this option.
You can also load coupons to your loyalty cards from coupons.com, pgsaver.com or smartsource.com
Along the lines of this is SavingStar which is an app that populates a handful of coupons monthly. Go into the app and select the coupons you'd like to use and it loads onto your stores loyalty card. When you swipe your card at checkout, you will receive a rebate or reward based on what was listed and selected in the app.
Rebate/Reward Coupons: I'm not really sure what the couponing professionals call this, but these are typically incentives, rewards or rebates that you get at checkout. It does not deduct money from your checkout balance, but it does give you a reward or rebate for the applicable products you purchased. Savings Star and UPROMISE are an example of this.
Did I forget any? Let me know in the comments!