To pick locks (lockpicking) in the Elder Scrolls Online you will need lockpick(s) which can be found in your travels or purchased from select merchants. I have never needed to purchase these though (at least not early on); as the game gets harder that likely will change. With the lockpick(s) you will be able to open locked chests (and so forth) you find in your travels.

When you find a locked chest for example you will be able to initiate the lockpicking action. The difficulty such as 'Easy' will be displayed; so you will have a good idea of what you are getting into. Once the lockpicking action is initiated you have as many attempts at picking the lock as you have lockpicks. Failure will result in the destruction of a lockpick. Having started you will see a cut-away of the lock showing your lock pick and a number of pins. Each pin will need to be set; setting a pin involves pressing the left mouse button which will depress the pin and then releasing the left mouse button at the right time to set it. Timing is of the essence.

There is a timer for the lockpicking process; if the timer runs out you lockpick breaks. While trying to set a pin, holding the left mouse button, you will need to listen to the noises made during the process, and also watch carefully the way the pin reacts; these cues are critical to the timing process in setting the locks. If you fail in setting a lock your lockpick may break. You will need to set all of the pins in the time allotted.

This video provides a good overview:
https://www.youtube.com/watch?v=j9o8Xak1_pM