There really isn't any one place to get all the information you need for this job, so here it is. Replacing a chain is easy and it's like riding a new bike. Very smooth pedaling. Highly recommended. It's a 15 minute job.
First, count the number of sprockets on the back wheel. Sprockets are the gears. For me it was 7.
That's the number you need to order. This is a 6/7/8 chain, so it will work if you have 6, 7 or 8 sprockets.
You'll need a chain tool for about $12. I got the IceToolz Chain Tool 7-10 Speed Shimano HG/IG/UG chain tool, but note that the 7-9 size costs about half as much. Your chain tool should also have a number that matches any number on the chain (not the number of sprockets). My tool says 7-10, so with 7 as one of the sizes of the chain I'm good to go. Note that even if you have 6 sprockets, because this chain *could* fit 7, the 7-9 or 7-10 tool will work for a 6/7/8 chain. It would not work for a 5/6 chain because 5 and 6 are not within the 7-9 or 7-10 range of the tools.
A chain tool lets you break links and put links on. There is no such thing as a master link: you just use the links that are on the chain to connect to one another. One reviewer said he used a missing link - you DON'T need a missing link or other kind of master link. The only thing a missing link will do is it will let you put the chain on without a tool, but putting the chain on with the tool is easy if you use the trick I describe below, so I'd skip the link and just buy the tool. If your chain is broke, you can use the missing link if you really want to otherwise, you'll need the tool to pull the chain off, so you can use it to connect the chain without a missing link. Connecting the chain is trivial.
You might also want disposable gloves. The new chain is greasy.
Shift to the smallest diameter sprocket, front and back. Note where the chain goes: it's usually over the top derailer sprocket and then under the bottom one.
Now, here's the difference between a 15 minute easy job and a half hour of problems: pull the rear derailer all the way forward and use a wire or cable tie to tie it to the bike frame.
The chain links are connected by pins. Pick any one on the old chain and use the tool to push it through. Pull the chain off.
Hold the old chain up to the new one with the top ends matched and find the link on the new one that matches the last link on the old one. Be careful: a very old chain will have stretched a bit, and so you may have to match the links halfway down the chain so that you get the same *number of links*, not the same *length* chain of a stretched chain.
Hold what should be the last link of the new chain with your fingers and set down the old chain. Push the lower pin on the link you are holding completely out of the new chain. The lower pin is the one that connects the link you are holding with the next one, the first one you don't need.
String the end of the new chain through the sprocket and derailer, starting with the end that has no pin. The other end has the pin sticking out of it so that you can push it right back in to link it, but with that pin sticking out, you can't thread that side through, so use the side with no pin to thread it through the derailers. If you can, you want the pin sticking out on the other end to be facing away from the bike, but it's not fatal if it's not. Note that the chain will be threaded *through* the bike frame: you can only string it through before you connect it: you can't connect it and then get it on the bike somehow. The two ends of the chain should be hanging down.
Match up the links and place them together. If they won't go together, separate the open end a bit with your fingers, and then slide the open end over the midpoint of the link on the other end, then pull it back to the proper position so that the ends line up.
Use the chain tool to push the pin sticking out back in fully. The two end links should swing somewhat freely (though not as much as the others): you may have to push the pin in a little more or push it out from the other side. Now work the link up and down and side to side to get it to swing a bit more freely.
Undo the wire or cable tie you used to hold the derailer forward against the bike frame.
Get a chain lubricant and lube the chain. The grease on the chain will last a few miles: it's primarily a rust inhibitor. I use Tri Flow oil, but I'm not a super serious biker. More serious bikers will want something pricier.
Ride about a mile to let the oil work its way in to the nooks and crannies of your chain, then take a paper towel or rag and wipe the excess oil off. This will keep the road dirt from attaching to the oil, gunking up the chain and grinding it down.
The "new bike feel" really comes from having a new chain (it's very smooth to pedal) and a fully adjusted set of brakes with new pads. I changed the pads, and adjusted my brakes, and a used bike I bought with a rusted chain really felt like new.
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Shimano CN-HG50 6/7/8-Speed Chain, Black.
