In the Buffyverse a Hellmouth is an area of weakness between dimentional barriers, literally a mouth into Hell (one of several Hell or Demon dimenions). The mystic energy radiating from the Hellmouth attracts demons and other supernatural creatures. The irony of Sunndydale as a name is intentional, as it was founded by a human who somehow gained the ability not to age, and who was striving to ascend to a full demon form (a lot of season three is based on this concept). At the end of the series we learned that Sunnydale was only the major Hellmouth, with several smaller ones throughout the world. The next largest one is located in Cleveland, Ohio.
In Buffy, a vampire is considered an animated corpse that has all the memories of the human, but its body is inhabited by a demon. There are two known exceptions with the characters of Angel/Angelus and Spike. Angelus was cursed with a soul by a Gypsey tribe after he killed a young Gypsey girl (the souled vampire is known as Angel), and Spike sought out his soul because he was in love with Buffy the Vampire Slayer.
The typical weapons such as stakes (or any piece of sharp wood inprovised from broken furniture, pool sticks, etc), Holy Water, decapitation, religious symbols, and garlic are effective (though according to Spike the garlic isn't really so as he would eat spicey and garlicky buffalo wings. In all cases, a vampire turns to dust when it dies. If a vampire is tricked into drinking Holy water it will dust (Buffy season three episode "Helpless").
As for the people not knowing what was going on, it was a kind of running joke in Buffy that everyone was oblivious, except for a few individuals. The people would shrug off or ignore what can't be explained.
I'm not sure how legal
this is, but it's a site where you can stream tv show episodes - they have quite a few Buffy episodes for anyone interested.
