There's several ways to make coffee that predate the coffee machine. The familiar drip machine is a development of the chorreador, basically just a cloth bag to hold the grounds. Modernized versions (basically just a filter holder that fits over your cup) are available, like this one.

Another method is to cold brew the coffee - essentially soak coarse coffee in cold water and strain through a cloth. This is excellent, by the way.

Then there's a whole slew of methods variously called "cowboy coffee," "camp coffee," "hobo coffee" and so on, which (typically) just involve simmering the grounds in water and allowing the grounds to settle, similar to Turkish coffee methods, which of course pre-date coffee makers as well.

One of the oldest ways to consume coffee is "Turkish coffee" which is nowadays included in one of the the cultural heritage lists by UNESCO.

It consists basically on mixing water, a very fine powder of coffee and sugar -optional- in a small pot and heat up. Pour on a cup and drink.

You can just dump coffee in a mug, pour boiling water on it and wait till it settles. Kopi Tubruk is traditionally drunk in Indonesia, in Poland boiling water is also added to ground coffee. Turkish coffee is usually made from cold water with ground coffee added and boiled. Even the good old cowboys made their coffee that way. Personally I have learned to drink it the Polish way, the easiest and fastest way of getting a good mug. One table spoon in a dry mug, add boiling water, wait a minute for the coffee grind to sink to the bottom and it is ready to drink. Watch out for that last sip, or you will get "spitting coffee" Sorry for the bad sources: http://www.talkaboutcoffee.com/types-of-coffee-coffee-varieties-II.html http://www.specialty-coffee-advisor.com/cowboy-coffee.html http://matadornetwork.com/life/ways-brew-coffee-world/ etcetera. Google for Turkish Coffee and you will find a lot of examples

If I may catch a ride on this thread how were these pots, popular until the 70's, used?
http://www.aroominmyhouse.co.uk/vintage_retro_home_ceramics-meakin_inca_coffee_pot_files/CR00-Inca%20Coffee%20Pot-a.jpg