How do I make redstone repeaters work?

The first thing you want to do is make sure that the repeater is in range. A single redstone current travels 15 blocks, so as long as the repeater is placed within that range, it will work. Additionally, regardless of where the repeater is placed, it will emit a signal strength of 15 blocks.

Secondly, repeaters, when placed, have a "direction". This is indicated by an arrow on the plate:

If a repeater is placed in a direction that does not follow the direction of the redstone it cannot repeat the signal:

1. First, make sure that the repeater is facing a wire (place the wire by putting redstone on the ground)
2. Then place the repeater in the correct direction. (example: if you put a wire facing east, put the repeater from east or west to work)
3. Check that you have a power source. If all else fails, make sure the whole wire is powered. Do this by powering the start and making sure the entire line is glowing. If not,try placing a repeater at the least glowing position of the wire.