Humans are fairly unique in that we posses the enzyme AMYLASE in our saliva. The more traditional name is Ptyalin. The only other groups of animals that have it are rodents, rabbits, and primates (which we are). All mammals have pancreatic amylase. It's main function is to break down starches into simple sugars. Remember that starches are chains of sugars stuck together. Another name for them is complex carbohydrates.
The really unique thing about human salivary amylase is that we all posses more than one copy of the genes to make it. The strangest part of this uniqueness is that some people have a few copies, and other have many! It ranges from two copies to fifteen. The evolutionary advantage to salivary amylase seems to be related to aiding digestion of starches. It is thought that pre-human primates started eating roots and tubers that were full of starch. Clearly the presence of extra amylase would help to break down the starches more quickly and the energy contained therein would be extracted that much sooner. Studies show that populations that have traditionally had higher starch diets have more amylase genes (Nat Genet 2007 Oct. 39(10) 1256-1260).
The other evolutionary advantage of higher salivary amylase levels is related to our insulin responses. If you have a greater amount of salivary amylase, you make sugar out of the starch you just consumed. That sugar triggers the pancreas to notice the sugar and secrete insulin sooner than it would have without the action of the salivary amylase. The insulin starts to help move the sugars out of the blood to areas where they need to go. The result is that the blood sugar levels never get quite as high. This is a huge advantage, as higher blood sugar is linked to many chronic ailments. Think of the poorer health outcomes of diabetics and you get the picture. Chris Masterjohn explains it all very well at his 2012 Ancestral Health Symposium talk.
Another issue that arises from less salivary amylase is obesity and higher body mass indices. The reason may be that blood sugar spikes are dealt with by getting the sugar out of the blood as quickly as possible by the action of insulin. The spike is dealt with by storage of the sugar as fat (Nature Genetics 42, 492-497).
There is a role that salivary amylase plays in plaque formation. Oral germs like to stick together in colonies and adhere to teeth. Teeth have a layer of organic material on them called the pellicle. The pellicle is known to contain amylase. Orstavik and Kraus found in 1973 that certain bacteria have receptors for amyalse that help them bind to the teeth. I suspect that given ancestrally appropriate diets, plaque is not bad at all. Indeed, our colons have lots of bacteria in them (pounds worth!) and under normal conditions, we are disease free. Actually, they help digest things we can't, and also keep more harmful germs away. So plaque would seem to do similar things under the right conditions. Weston A. Price proved that traditional diets don't cause tooth decay.
Salivary amylase only becomes a problem when we move away from eating real food and consume processed carbohydrates. It is known that there is a correlation between the amount of salivary amylase we have and tooth decay (J Dent Res. 1950 Apr. 29(2) 165-172). This makes sense. We all know that sugar can lead to tooth decay. The more amylase one has, the more sugar they make out of starch. The interesting thing is that bacteria also make amylase. They like sugar! It was determined that bacteria hijack our amylase to help them make get more sugar from the starch we consume (Douglas, 1990; Scannapieco et al 1990). The waste that is produced is called lactic acid. It is actually the acid that dissolves the minerals out of the teeth, eventually leading to a cavity.
As an interesting side note, tea seems to lower amylase production from an oral germ called S. Mutans, one of the causers of tooth decay. It appears that it is the tanic acid that is responsible for most of the effect (Arch Oral Biol. 1988; 37(11): 845-6).
Amylase in our saliva is a good thing. Modern diets have changed more quickly than our ability to genetically adapt, making it a potential cause of tooth decay. I don't suggest that you drink endless amounts of tea as a preventative, rather stick to a more ancestral diet to maintain the environment of your mouth in a way more consistent with our evolutionary background.