Awakenings are passive bonuses which certain monsters can obtain on top of their leader skill/active skill. These passive bonuses are always active if a monster is on your team, regardless of whether the monster is a leader or a sub team member.
The bonuses range from stat boosts (which stack with plus eggs) to abilities such as canceling binds by matching a full horizontal line of hearts.
For a full list of monsters with awakenings, refer to this page: Monsters with Awoken skills - PAD wiki
You can awaken a monster once by fusing a Tamadra to it. Make sure your monster can actually awaken before wasting a Tamadra! You can also fuse a duplicate monster at the same evolution level but this is mostly limited to super whales.
For more info, such as how to obtain Tamadra, watch this video. Don't worry, it's short! Awakening System Guide
Here's another video on Awoken Skills: Awakening System Guide! Part Two: Awaken Harder
Help, I'm new and I got one of those Tamadra guys. Who should I use it on?
A general rule of thumb is that your main leader god, the one you leave up most of the time, is a good candidate to receive your first Tamadras. Other players are more likely to use your leader the higher its level, plus count and awoken count, and this results in more Pal points for you! Tamadras can (rarely) drop from the Pal machine so most players will benefit from this in the long run.
Ask in the thread if you have absolutely no idea which monster to awaken with your Tamadras. The answer to this question can differ considerably depending on the contents of your box.