There are two types of coupons...store coupons (these are coupons specific to that store) and manufacturer coupons (coupons sent out from the manufacturers that you can use at any store). When buying an item you can use ONE store coupon and ONE manufacturer coupon per item. This is called "stacking coupons". You can tell what kind of coupon you have by looking in the top left of the coupon. It will specifically say "Manufacturer Coupon" or " (store name) Coupon".
Some stores accept competitor's coupons, meaning you can use another store's coupon at their store. Most stores will have a list of competitors they'll accept, and some stores don't accept any. You have to check with the store to be sure. When using a competitor's coupon, it counts as a "store coupon", so you can pair it with a manufacturer coupon. You cannot use a store coupon, manufacturer coupon AND competitor coupon. Only a manufacturer and store OR manufacturer and competitor. I hope that makes sense. :)
I love the way Jenny over at Southern Savers explains it. She's got pictures and everything! (for us visual learners!)
I hope this helped clarify a bit. Are there any other questions about store and manufacturer coupons you have? Leave it in the comments section!