When I was at uni, the oral medicine course was taught by a pair of characters who were insanely passionate about their jobs. I will never be able to shake the image of their heads simultaneously trying to explode whenever someone used the term "blood thinners". It's a commonly used term colloquially, but it's not accurate, (and therefore the cause of much discomfort for those two academics obsessed with being technically correct).
Anticoagulants is the correct term, as these drugs don't cause the blood to be physically thinner, but rather they interrupt the various ways in which a clot is formed. Common ones are aspirin, warfarin, and clopidogrel to name just a few - and there are new ones coming on the market all the time. If you cut yourself whilst on anticoagulants, your blood will not flow out faster or appear less viscous, it will just take longer to clot. Anticoagulants are prescribed by medical doctors, (not dentists), to prevent unwanted clotting or to break up clots which have already formed. Where don't you want a blood clot? Pretty much anywhere inside your veins and arteries. Clotting is great for stopping bleeding, not so great when all your blood is still inside your circulatory system where it should be.
So what happens if a clot forms where it shouldn't? You get things like deep vein thrombosis, heart attacks, strokes. Not good! If a piece of a clot breaks off and travels into the smaller blood vessels of the heart, lungs, or brain, it will block off the blood flow. That's why people at risk of developing clots take anticoagulants - there are some pretty serious consequences!
Enough boring background, what does all this have to do with dentistry? For a long time it was recommended that people taking anticoagulant drugs stop them before having a tooth taken out, because of the risk of excess bleeding. This was really convenient for the dentist, but not so for the patient. It can take several days to weeks of being off medication before clotting starts to return to normal, which means there is a significant amount of time in which the patient is at risk of an unwanted clot forming - all so a tooth can be removed with minimal bleeding afterwards. Looking at the big picture, patients are in more danger from having a dangerous clot form during this time than they are from bleeding after a tooth extraction. That's why in recent years the guidelines for dentists have changed, and it's no longer recommended to stop anticoagulant medication for oral surgery.
Letting people bleed isn't a viable option either, so instead your dentist will take a number of precautions that don't interfere with the potentially life-saving anticoagulant medication. These include having your clotting status checked via blood test ON THE DAY of your surgery, placing stitches after the extraction, packing the fresh socket with a material that will aid clotting, and the use of a special mouthwash which further promotes the formation of a good clot.
Taking anticoagulant medication need not be a barrier to having oral surgery. Doctors prescribe it for very important reasons, and it's the job of your dentist to manage your oral health in a way which doesn't compromise your general health. It's so important to keep your dentist up to date with all the medications you are taking, even if you think it's not important or relevant.