Partial-mashing is an intermediate step between pure extract brewing and all-grain.  Some describe it as "training wheels" for all-grain.  The truth is that all-grain brewing is quite a bit more complicated than extract brewing, and it takes some time to get used to controlling the temperature and volume of the mash to effectively extract fermentable wort.  Partial-mashing is an excellent way to get some practice with the mashing process without requiring a significant investment in new equipment on top of what you already have for extract brewing - and it's also a great way to improve your beers.

What is mashing? 

"Mashing" is a term for the process of converting starches in grain to fermentable sugars through the use of enzymes naturally present in the grain.  The reason it requires some work is that the mash needs to be in the right temperature range for the enzymes to function.  A basic description of mashing would be that the grain is crushed and added to water at the correct temperature, and the temperature is maintained for a period of up to an hour to allow the enzymes to do their work.  Usually the process involves several stages, such as 'sparging' where after the conversion is complete, additional hot water is added to help rinse all of the sugars from the grain.

There is quite a bit of chemistry involved in the mashing process, and if you'd like to learn a lot more about it I suggest you read chapter 14 of How to Brew by John Palmer.  If you want to consider all the chemistry involved, the mashing process could be very complex, but for the purposes of this article I will discuss a very simple method, essentially just two steps, referred to as a 'single infusion' mash, and a 'batch' sparge. 

Single Infusion Mash: This simply means that hot water is added to the grains to reach the desired mash temperature and then it is held there for about an hour for the conversion to take place, before draining the resulting wort into the brew kettle.  In this step, you need to use the right amount of water, at the right temperature.  The ratio of water to grain is usually about 1.25 quarts of water per pound of grain.  The mash should be held at a temperature typically between 150-155F.  Where you are in that range matters also, because there are two enzymes at work.  The lower end of that range is more favorable to the beta amylase enzyme, which converts starches into maltose, your main fermentable sugar - this results in a dryer beer.  The higher end of that range favors the alpha amylase enzyme, which produces not just maltose but also other sugars, not all fermentable - this results in a sweeter beer. 

Batch Sparge: This means that the wort is drained from the mash, more hot water is added, this time reaching a target temperature of more like 160-165 degrees, it is allowed to sit for a few minutes, and then drained again.  This step is to rinse the remaining sugars from the grain. 

How can partial mashing improve your beer? 

There are a lot of different grains that are suitable for brewing.  One of the easiest steps you can take as an extract brewer to gain some more variety in your beers is to use steeping grains.  However, only some grains are suitable for steeping - these are grains that contribute color, flavor, body, or mouthfeel, but don't contribute fermentable sugars - examples are crystal malt, roasted barley, and many of the 'cara' malts.  But, the majority of grains do need to be mashed in order to convert their starches into fermentable sugars.  If you are able to mash your grains, you gain access to a larger variety of grains for brewing, giving you a lot more flexibility in building your recipes.  Bodensatz Brewing has a list of various grains , with an indication of which ones need to be mashed.

What equipment do you need, and how much does it cost?

You can do your first partial mash for very little investment.  The cheapest method is to put your grains in a nylon or muslin grain bag and mash them in your brew kettle, using careful temperature control on the stove top, or putting the whole kettle in the oven to maintain the temperature.  Alternatively, you can do the same in a small cooler - there is a great article in BYO Magazine describing this method.  

Personally, I chose to spend a little money and convert a small 2-gallon rubbermaid beverage cooler to a 'mash tun', by adding a stainless steel braid and a valve, to allow liquid to be drained from the grain.  This method is much closer to that used by many all-grain brewers, and it allows pretty easy heat control without a tremendous investment.  This has the added advantage that you can go buy the 5-gallon version of the cooler later on down the road, and transfer the same hardware over to it in minutes, increasing your grain capacity to allow for very large partial-mashes and even all-grain brews.

The Process:

I will describe the process that I use with my 2-gallon insulated mash tun.  

1. Add crushed grains to the mash tun - with mine I can do about 3-4 lbs of grain.
   
2. Heat the 'strike water'.  I personally use the BeerSmith software to calculate the volume of water to use, and the temperature to heat it to.
3. Add the water to the mash tun, and stir carefully to make sure the mash is thoroughly mixed.
4. Put the lid on the tun, insert thermometer, and make sure the target mash temp was reached.  At this point, I also throw a towel over the top to help insulate it better.
5. Wait for an hour.  One rather interesting thing you can do toward the end is try the iodine test.  Take a small sample of the wort, put it on a white plate or bowl, and add a drop of iodine (available at any drugstore).  If the iodine rapidly turns a dark blue/purple color, there are still unconverted starches in your wort.  If it remains its normal yellow-brown color, the starch conversion is complete.
6. Toward the end of the hour, heat up your sparge water.  Again, I use beersmith for this calculation.
   
7. Drain the wort into your brew kettle.  You can go ahead and start heating it while you do the next steps.
8. Add your sparge water to the mash tun, and let it sit for 5-10 minutes.
9. Drain into the kettle again.

At this point, you are done with the mashing process and you can continue brewing as you normally would - start the boil, add extract, hops, etc.

Fair warning, spent grains will get really disgusting if you throw them in the trash and leave them overnight, so you would be better off dumping them outdoors or at least taking the trash out immediately.