There are many, many shrines and temples all over Japan. The distinction between shrine and temple is important although they might look similar to the untrained eye: temples are Buddhist places of worship, while shrines belong to the Shinto religion.
Most shrines are somewhat similar to each other with regard to the available facilities and their usage. However, the process of worship is very different from Christian ceremonies - as far as I can tell, anyway. The picture below shows a typical set-up.
Right in front, there is a bowl with burning incense sticks. Behind that, you can see a wooden donation box. The colorful string right in front of the donation box is pulled to make some kind of noise, in most cases strike a bell. And finally, at the back, there is the inner part of the shrine, which often features some statues and flowers.
So what do people do when they come to pray at a shrine? Everybody follows the same procedure, so it seems to be fairly fixed. First, they will wave some of the incense towards them, so it probably has some kind of meaning (but I don't know which). Then, they put some money into the donation box - apparently you are expected to donate each time you pray at a shrine. Then they make some noise to attract the attention of the enshrined deity - most people ring the bell, but some also clap their hands. And finally, they stand in silence and pray.
People often line up to ring the bell and pray, and the first couple of times I joined the queue because I didn't know what they were queuing up for - usually there is something worthwhile at the end of queues in sightseeing spots. But by now, I have learned that I can safely skip the shrine queues ;-)