Another popular method is poured coffee, or the drip. when you pour water, rather slowly through some sort of filter that contains the coffee beans, you usually extract more of the flavour of the coffee beans, though you do give up some of the full-body texture that people appreciate in espresso drinks.
Another alternative is the French Press or the Percolator. Basically, the idea is the you use you arm’s strength to push the coarsely grounded coffee beans through water, thus creating something of a hybrid between a drip and espresso.
You also have another type of coffee which is a cold brew. In this type you actually mix the ground coffee with water in cold temperature (which according to some coffee mavens, hurts the coffee less, thus extracts the flavours better), and keep the water coffee together for anything between 8–24 hours, after which you run the mix through a filter or a sieve and get your cold-brew.
Another option for mixing coffee and water is actually with very hot (or nearly boiling, or boiling, depending on your ideology) water. To make Turkish Coffee, for example, you mix the coffee with the water, boil it (or not) a few times, and keep the ‘mud’ at the bottom of your coffee cup.
And there are quite a few others. People are making all kinds of devices for brewing coffee today. Run a quick search on kickstarter and you will find some really cool stuff.
And also, there is coffee that isn’t coffee at all, that all of people actually call coffee. That’s instant coffee for you. That used to be coffee which evaporated, heated and what you are now drinking are the extracts of that procedure. The taste is really different than coffee, but it does contain caffeine.