This is intended to explain why repeaters are often necessary.  It will also give you a basic understanding of how and what a repeater is.

To understand why repeaters are valuable to us in the first place let's start with a review of the most basic communication method.

In the first illustration we see that Paul and Susan are able to communicate directly.  Radio to radio.  This is the most basic, most reliable and cheapest form of radio communication there is.  Because of this it is the method preferred by all amateur and professional communicators.  Notice the straight double arrow between the two people?  This represents that there is no obstruction between them.  They have "Line of Sight" or "LOS".  This is required, for the most part, at frequencies in the VHF bands and higher.  that doesn't mean that the people have to be able to see each other, it simply means that there is a line between them that is free from obstruction.  When this situation exists the two people can talk reliably....within range.  A rough estimate of range is a mile or two per Watt of transmitter power.   So if Paul has a 25 watt transmitter and Susan is only a few miles away, they're talking.

What if Susan is farther away?  Suppose she is still within "Line of Sight" but she's 50 miles away from Paul.  If they upgrade to 50 or 100 Watt radios, and the power supplies to support them, they might still be able to talk.  But beyond that, she's going to be lucky to hear him at all.  Even if she does, it's doubtful that she'll be able to understand everything he says because there won't be enough signal to overcome the static and noise.  What they need here is some help.

Fortunately for them, Chuck is halfway between them and also has a 25 watt transmitter.  He can hear Paul very nicely.  And Susan can hear Chuck well.  So Paul sends his message to Chuck who writes it down.  When Paul is done speaking Chuck transmits to Susan and reads the message to her.  Mission accomplished.

Our national amateur radio organization in the US is the American Radio Relay League (ARRL).  Think about that name for a moment, really consider those two words in the middle.  Back in the day those words really meant something.  The ARRL got its start, and its name, as a group of operators who relayed messages exactly the way I just described.  It was the only way to extend range back in the early days and they developed methods for reliable and efficient traffic handling that are still used today.  (Those are skills you should have and hone).  

Imagine that Chuck has a life.  He doesn't mind helping Paul and Susan once in a while, but he has better things to do all day than stand around waiting for the opportunity to pass messages between them.  What if we could literally "cut out the middle man"?  We are trained radio operators who studied electronics in order to get our licenses.  Of course we can come up with something.   Chuck listens to Paul.  To do this he uses a receiver.   Chuck transmits to Susan and he uses a transmitter to do that.  If we could wire Chuck's receiver directly to his transmitter, we wouldn't need Chuck there standing around all day.

Chuck's done exactly that in the illustration above.  There is a problem with this however.   Because the receiver and transmitter are on the same frequency, the power from the transmitter is going to desensitize the receiver.  They are just too darn close.  Chuck didn't have this problem when he was in the loop because he would receive and write down the message and then he would transmit and read the message.  Chuck, like most of us, can't listen and talk at the same time.  There are ways, using recording devices, to do exactly what Chuck was doing.  But we can speed things up and pass the message in real time by simply putting the receiver and transmitter on separate frequencies.  Doing this allows the transmitter to be on without wiping out the receiver.

Here is the same thing again, this time with a 600 KHz difference between the receiver and the transmitter.  If you've been using the 2 meter band for any length of time, this will look familiar to you because 600 KHz is the standard split on that band.  In UHF (70 cm) the split is 5 MHz.  The farther apart they are in frequency, the less the transmitter interferes with the receiver.  There's more to this and it's something of an art.  But I think you've got the general idea here.  Chuck now has his receiver wired so that when it receives a signal it passes that signal to the transmitter....in real time. 

Here we see Paul transmitting his message on 144.630 MHz.  Chuck's receiver is picking that up and passing it to the transmitter which is putting it on the air at 145.230 MHz.  When Susan wants to reply they process is reversed with her transmitting on 144.630 MHz and Paul receiving on 145.230 MHz.  Chuck, relieved of his duties, has gone fishing.

Remember that the repeater is needed here because Paul and Susan are too far apart for the power they have.  That can be quite common, especially in places like Eastern Montana, Iowa, Kansas, Texas, etc.  That is not our challenge here in North Idaho.  Step outside and look around you, or fire up Google Earth for a moment.  Chances are you've got a radio with 25 Watts or more of power and an antenna with some gain.  So you should be able to talk, with line of site, about 25 to 50 miles or so before you exhaust your the ability of your power.  With that in mind, what's in your line of site?  Mountains most likely.  And how close are they?  Well within 25 to 50 miles in our area.  The point is this:

A 25 Watt mobile radio in North Idaho will perform as well (or better) than a radio with more power.

Here's another way to say it, increasing your power beyond 25 Watts will do little or nothing for you.  Long before your 25 Watt signal runs out of oomph, it runs into a mountain or other obstruction.  Where Paul and Susan could have solved their previous problem with more transmitter power, to a point, that just isn't going to do a damn thing for us here in the mountains.  In fact, increasing transmitter power will often reduce communication range, particularly in the UHF band and higher due to multipath interference.

But we can apply the same solution as before.  Although the mountain is our obstruction, we take advantage of the fact that our signals are reaching the mountain, usually with plenty of power to spare.  We take Chuck's repeater and place it on top of that mountain, Paul and Susan are able to communicate again.