There are two ways to start with: plunging and dripping.
The slowly / "much slower" part means about 8-12 hours, depending on the quantity of ground. For example, I use 400mL/14oz and 40g/1.5oz of ground on an 8 hour drip.
The beans should be quite finely ground, as cold water does not penetrate the ground as well as hot water.
@henryJ exposed plunging in detail. As for dripping, it is possible to use a standard filter stand and paper filter.
The hard part is to drip water slowly. There are cheap devices on the market. An alternative is to get one of those flasks used in chemistry class (a new one, never try with a used one---chemistry is scary). The cheapest way is probably to get an empty plastic bottle, put the cap on, use a needle to make a tiny hole in the cap, fix the bottle upside down above the ground to start dripping. The goal is for the water to pour drop by drop, at a rate of about 1 drop every couple seconds.
Cold brew is simply brewing coffee in cold water. It's usually a long process (about 12 hours), but totally worth it. The simplest method is using a french press. Measure the amount of coffee and water you need for a regular hot french press brew, but instead use cold or room temperature water. Leave the press pot in a refrigerator overnight or at room temperature, push the plunger down in the morning, and you'll have delicious cold brew ready.
You can also buy cold brew systems. I've never used it, but I heard this works pretty well.
Excellent description and answers already (upvotes all around...); here's another take and some links to external resources and experimentation.
I use the "plunging" method, mixing grounds directly with water, as described in other answers by @henryJ and @Eric. I find that the outcome of brewing, then filtering via French press, is too gritty or grainy, far more so than the store-bought stuff (which might be slow-drip, but I don't know how it's done on a production scale).
Filtering that result through a conventional coffee filter (e.g., like this) takes a long time -- on the order of hours for some reasonable quantity of coffee. I don't know exactly why, but I attribute this to the longer brew resulting in more stuff that clogs the paper filter; this clogging seems to happen no matter how coarsely I grind the coffee. (I'd be happy to hear suggestions about that, though; doesn't seem enough for a question...)
I use a multi-step filtering process, putting the coffee through a few passes of cheesecloth or other cloth to hopefully filter out the bigger chunks first, but this is messy (hard to clean up) and seems wasteful.
Here's a few other techniques with similarly absurd steps to filter the coffee during or after brewing. Both are steeping in the style of a tea bag, with the grounds wrapped in a sack.