How do you replace the chain? You need a chain break tool. Many of the multi-tools that are carried on the bike have one, but they also make single-purpose models. The chain is assembled with a pin press fit into the plates. The pin is set between four plates (two inner and two outer). There is also a spacer between the inner plates. To take the old chain off, you use the tool to push the pin out through 3 of the 4 plates. Note that some chains have a "power link", that allows chain removal without the use of a tool. This link is sometimes a different color (gold or silver) for easy identification. To remove the "power link", the links on either side must be pushed together until the links unsnap. I usually just find it easier to push the pin out.
Take the old chain off and stretch it out on the workbench. Take the new chain out and stretch it out alongside. You'll probably find the new chain is longer. You'll need to remove a few links to make the new chain the same length. Always remove an even number of links unless you have a power link (then remove an odd number). Trust me on this one.
Put the new chain on and you're ready to go. If you are not familiar with how a rear derailleur works, you might want to take a picture of it prior to removing the chain as the new chain must thread back through the derailleur the same way. Use either the chain break tool to push the pin back in or use the "quick-link" that came with your new chain. You may find the chain easier to put on if you shift both the front and rear derailleurs to the smallest cogs or chainrings. This will give you extra slack to connect the ends.
In the next part of this series, I'll discuss the most controversial topic in the universe, more contentious than politics and religion combined: CHAIN LUBE!