The Japanese suffixes have been explained many times in the net, but so you won't have to look for them, here's again one for you. There's also a quite good explanation of some of these suffixes in the beginning of Tsubasa, published by Del Rey, if you're interested. It's a good manga, too.
The suffixes are added at the end of the name, like Suzumiya-san, Kaori-chan or Inue-kun. In addition to suffixes, also the choice between first and last name is an important factor, first name of course being closer. The Japanese talk to each other with their last names, unless they're specifically invited to use the first name.

-san
The most common suffix, used in any kind of situation that requires basic respect for the other person. San is used for both men and women (and girls and boys). This is for example used when talking to your neighbours, people you've just met, or not so close friends. In school students call their female classmates with san. (Male classmates are called -kun, see below.)

-kun
People usually hear this suffix in the school scene. Students use kun for their male classmates. It stands for a little less respect than san, giving the impression that the speaker has a little higher status than the person spoken to. (Girls deserve more respect than boys in school...) It is also used among friends, and then the status thing doesn't matter that much. Kun is usually associated with the male gender, but it can also be used for women in certain situations. One of these could be an employer talking to his employees.

-plain name
Much more than in the western countries, addressing someone with their plain name gives the impression of a close relationship. Plain surname is not that much closer than kun, but using plain first name means a really close relation between the speakers. Plain first name is mostly used by family members, long-time friends or lovers.

-chan
This suffix is used mostly for children and between girls that have been friends for a long time. It's something vaguely like "dear" or "sweetie". For a child growing up, mostly boys, being called chan could be very embarassing. Between girls it just means they hold each other very dear. Chan is always used with first name, unless you want to be a bit funny. Chan is also used for pets and the like.

-sensei
Sensei basically means "teacher", but also "doctor" and a "master of some art". There are some other situations where you could use it, like maybe when talking to the head of the office, but "teacher" is, I think, the best translation. Sensei of course goes for almost any sort of a teacher, not just school.

-senpai
This is a suffix also associated with school life, even though it has other uses. A sempai is someone who is "ahead of you with his studies", meaning an upperclassman or a more experienced student. It is also used in working life for employees who have been in the office longer than you.

-kohai
Kohai is the opposite of senpai. It's not used as a suffix that often, though, since people tend to use san or kun to replace it.

-sama
Sama is used for royalty and people of really high status. Also, you can hear service personnel using it when talking to their customers. In anime you can hear it used as a means to flatter or to lift the other on a pedestral... Basically this is the formal suffix for extreme respect.

-dono
This suffix isn't used that much, since it's somewhat old. You may hear it in stories, though, and maybe from an old and traditional person. It's been used as both higher and lower than sama depending on the era and the region, but it always carries a great respect towards the person spoken to.

-neesan, -niisan
These suffixes literally mean "big sister" and "big brother". They're maybe not real suffixes, but children use them a lot, both of their own siblings and of other suitably aged people.