Switching downspouts to the other end of a gutter
You will need:

1 - Rivets or screws. I would recommend rivets for aluminum gutter

2 - An outlet for the new downspout, if you can't reuse the old one

3 - A piece of metal slightly bigger than the width of your downspout

4 - Aviation snips, if you currently don't own any

5 - Gutter sealer or solder bars

6 - A new downspout, if needed

7 - As tester101 mentioned, you might need to change the slope of the gutter.

Now, All gutters should be sloped, but, this is not always done and is not always that big of a deal if the gutters is not that long. However, if it is sloped already, then you will most definitely need to remove the hangers and change the slope from one side to the other.

Directions:

1 - Start by removing the old downspout, however attached it is. If soldered you will need a torch or a soldering iron. If not, rivets or screws are easy to remove.

2 - Remove the outlet.

3 - Patch the hole with a piece of sheet metal, copper aluminum or steel

4 - If copper, solder the patched metal for that you will also need an soldering iron. To solder the copper together you also need soldering fluid, or acid. A torch will heat the metal too much and you won't be able to solder it together. If aluminum or steel, rivet it together and seal it with gutter sealer.

5 - Adjust gutter slope if necessary

6 - Using the newly purchased or older outlet, cut the new hole for it on the other side of the gutter. Use the outlet as a jig. Place over the bottom side of the gutter, as it will be easier to cut the hole. Using a scratch all or sharpie, mark the outlet on the gutter. Make sure you mark the INSIDE of the outlet.

7 - Place new outlet. Example

8 - Solder it if copper, or rivet it and seal it if aluminum or steel.

9 - Attach the downspout.

This should be done at your own risk. There's a reason professionals do it daily. 

Have fun!