How do I become a Werewolf?

I've heard that one of (maybe the only?) secret races in this game is a werewolf.

How do I become a werewolf?

And by extension, how do I cure lycanthropy, that is, if there even is a cure?

The upper Echelon of The Companions (Fighter's Guild), called The Circle, is composed of Werewolves. As you advance through the Companions quest line, you will eventually be inducted into The Circle, whereupon they will perform a ritual to turn you into a werewolf.

(You need to become a werewolf to continue the Companions questline, but you aren't forced to complete the quest (and become a werewolf) if you don't want to).

And by extension, how do I cure lycanthropy, that is, if there even is a cure?

Fortunately, you are also introduced to the cure when you are doing the Companions quest line. You'll be asked to kill some Glenmoril Witches by Kodlak, and make sure to hang onto their heads. The questline will tell you exactly how to cure your Lycanthropy.

Source: Cthos

Raven Dreamer's answer doesn't cover the second part of the question:

And by extension, how do I cure lycanthropy, that is, if there even is a cure?

Fortunately, you are also introduced to the cure when you are doing the Companions quest line. You'll be asked to kill some Glenmoril Witches by Kodlak, and make sure to hang onto their heads. The questline will tell you exactly how to cure your Lycanthropy.

Dawnguard DLC

Without the Dawnguard DLC, you can only become a werewolf once. With the Dawnguard DLC, you can become a werewolf twice, if you've cured yourself of Lycanthropy the first time around.

According to the UESP wiki's "Aela the Huntress" article:

If you have cured yourself of lycanthropy, there is no way to regain it in the base game. However, the Dawnguard add-on provides a one-time opportunity to restore your lycanthropy by speaking to Aela.

After being cured of lycanthropy, simply find Aela and speak with her. You will have the option to tell her you wish to regain the gift of beast blood. She will respond, "Do you, now? Hircine doesn't usually bestow his favor on the fickle. But you've proven your worth. I'm willing to try. Are you ready now?" If you tell her you need more time to prepare yourself, she will end the conversation by answering, "Go. Reconsider your plight." If you then return to her you can go through the conversation again, and it will continue as before. When you choose to tell her you are ready, she simply says "Hold still, then." Your screen will turn black for a second, then a magical aura will circle around you and a message will appear on the top left of your screen stating "Beast Blood Added". The conversation is now over, and if you choose to cure your lycanthropy again, there will not be another chance for Aela to restore it.

how do I cure lycanthropy, that is, if there even is a cure?

With the Dawnguard DLC, you can cure Lycanthropy by becoming a Vampire Lord, either through Serana or Lord Harkon (details in the linked UESP wiki article). You can then choose to cure yourself of vampirism.

Console commands

If you are playing on the PC and have the Dawnguard DLC installed, you can use console commands that will allow Aela to turn you into a werewolf after your second time to be turned (allowing you to easily switch to being a non-werewolf and a werewolf back and forth as many times as you would like). You can do this by entering the console commands then conversing with Aela to turn you into a werewolf, and conversing with Serana to turn you into a Vampire Lord (or just be a vanilla vampire), and then cure yourself of vampirism if you like.

You can enable the conversation option "I wish to regain the gift of beast blood" to Aela with the console command: resetquest 0200f899

If you encounter issues with the above console command, enter player.setstage F6090 100 to force a cure before using the console command to enable the conversation option.

^{Console commands info from: Nexus Mods forums post, "Dawnguard Regifting Lycanthropy", by user, speedyfox}