/r/askscience might be a better place to ask this since they can describe it to you in more detail, but I wager many of us, through our situation, know enough about sailing to give you a general idea of how it works.

The process is called 'tacking'.

It's true that if you only had a sail, you'd be only able to travel which ever way the wind blows. Such is the situation with hot air balloons which have to change altitude in order to find different wind currents. However, wind is not the only tool at your disposal; you also have the friction of the water against your boat. Keels can help greatly in this.

A square sail aligned perfectly perpendicular to the boat would not be able to sail against the wind. But if you can turn the mast, even a square sail can be used to sail 'against the wind' with some success.

The truth is, sailing directly against the wind is impossible. What sailors do instead is sail almost against the wind. By sailing at an angle to the wind, and having the sail itself angled somewhat toward it, a few things will happen. The air will blow on the sails, but friction against the water will mostly prevent the boat from traveling in that direction. The wind will be deflected off the sail at an angle parallel to the ship, where through simple Newtonian mechanics, imparts momentum that propels the ship forward. A triangular sail will minimize drag faced by the ship, making it faster (though it may have some other advantages I'm not aware of).

Think of a sail like a mirror, with wind being a flashlight (or laser pointer, if you want to better see where the light is going). Shining the flashlight directly in front of the mirror results in light being reflected directly back. Now shine the flashlight directly behind the mirror, and angle the mirror until the light is deflected away from the original light source. A lot of this can actually translate directly to how real life solar sails work, too.

As a result, ships don't usually sail directly 'against the wind', rather, they zig-zag toward it. It's not as fast as sailing along the wind, but it sure beats rowing.

Technically, it's not possible to sail directly against the wind. Instead, ships would angle slightly left or right of the headwind, and then utilize a zig-zagging motion to head upwind. This technique is called 'beating upwind'

It's important to note also that the rigging type of a ship determined it's capacity to sail against the wind. A lateen sailed ship has a far better ability to sail upwind, whereas a square rigged ship would have a tough time, and would need to take a wider angle when beating upwind.