The following actions are considered crimes and are reported if you are detected:
-Killing chickens or other farm animals owned by people
-Stealing items or horses
-Interfering with official empire business
-Not following guards' orders
-Picking a Lock
-Starting a fight (defending yourself is not a crime)
-Killing a non aggressive character
When a crime is reported, a Bounty is placed on your head for the hold you are in. The amount of that bounty depends on how serious the crime is. Anyone who sees you commit the crime is a witness. If you manage to eliminate all of the witnesses to a crime shortly after it has been committed, the bounty will not take effect. However if in attempting to get rid of the witnesses, you commit more crimes that are witnessed by additional people, the bounty will rise. Note that if you are going to kill the witnesses, you should do it as soon as possible, otherwise some of them would have already reported your crime to the guard.
When you have a bounty on your head, guards will attempt to approach and arrest you. If you flee, they will chase you. If you manage to escape to a different hold where you do not have a bounty, the guards there will ignore you.
Be careful with your weapons and spells in town. It is easy to accidentally cast a ranged spell and hit someone with it, and get a bounty over your head as well as turn the people in town to hostile towards you.
Brawling is not considered a crime unless you decide to draw a weapon or cast a spell.
Vampirism and Lycanthropy (Becoming a Werewolf) are also considered a crime.
If caught by the guards you typically have the following options:
Pay your bounty. Any stolen items in your inventory will be confiscated, including items you stole undetected, but you are otherwise free to go. Usually this option gives you the least trouble. But you lose gold on the fines.
Go to Jail. After being sent to jail, you can attempt to pick the lock of your jail cell and escape. Although the guards take all your items, you manage to sneak a picklock into your cell. Use it carefully, since you only have one chance to break the lock. If you manage to escape, get your belongings before you sneak out.To serve you time, activate the bed in your cell and you will appear outside when your time is served. You will lose progress towards skill increases depending on how long your sentence is.
Resist Arrest. Guards will attack you if you resist. If you escape, the Bounty will remain on your head, and guards will try to arrest you if you return. This can be dangerous and the bounty can rise as you assault more guards in the process. If you yield by sheathing your weapon, the guards may stop attacking, but they will still attempt to arrest you.
If you are the Thane of a town, guards will agree once to look the other way. This won't work though if you continue to commit crimes.
If you have the Bribery perk (under speech skill tree) you can sometimes bribe guards to ignore your crimes.
In the main menu, under "General Stats" you can view the stats for your criminal activities:
Total Lifetime Bounty
Spread The Word
Skyrim Crime and Jail Guide