Bikes are machines, and any machine will invariably need maintenance at some point in its life.  Of course, you can have a bike shop do the work for you, but you should always have some tools handy in case they’re closed, or you need to do some emergency repairs while you’re out riding.  The recommended tools are listed below, divided into categories.

Tools to have on the road:

1. A multitool.  Sort of like a Swiss Army Knife for bicycles, this tool can save your bacon when a screw comes loose in the middle of the countryside.  Make sure you choose one with a built-in chain tool.  Always have it with you.
2. A patch kit.  Some patch kits are as small as a digital memory card and very light.  If your tire is punctured, a patch kit is a great alternative (though not necessarily a guaranteed replacement) for a spare tube.
3. A small hand pump or CO2 inflator.  What good is a patched tube if you can’t reinflate it?  I recommend the Topeak Road Morph pump, for high pressure tires.
4. Tire levers.  Cheap and compact, tire levers are a necessity if you need to fix a flat tire.
5. Zip ties.  These will temporarily hold together pieces of your bike that have fallen off. 
6. A cell phone, or a few quarters for a pay phone.  If something really catastrophic happens (and I really hope it doesn’t), then you’ll need to call someone to come pick you up.

Tools to have at home:

1. A full set of metric hex wrenches.  Sure, you can use a multitool for all your work, but it’s easier to use dedicated tools.
2. A chain tool.  A dedicated chain tool is easier to use than the one on a multitool.
3. A floor pump.  Much easier to inflate a tire with one of these!
4. A repair stand.  Great for tuning your derailleurs, and for storage if you don’t have a kick stand.
5. A chain cleaner and brush.  When your chain gets covered in road gunk (and it will), these are your friends.

If you’re really serious about home repair (in addition to what’s listed above):

1. (more info coming soon!)

For loads of great information on bicycle maintenance and repair, check out these sites:

Park Tools

Sheldon Brown's excellent site on all things Bicycle