Currently a b'attle' commences when you attack map squares occupied by Barbarians and Players, or defend your cities from a Barbarian Raid. In the pre-release, there are no options for attacking player settlements. After a battle, a "battle report" appears in your activity stream that shows the troop numbers of engaging forces and losses for both armies.
Attacking
To engage in offensive battles outside of your city, you need military units and the Parade Ground building. The higher the level of the Parade Ground, the more armies you can send out - this includes occupying armies and is initially 2, but increases later on.
There is a difference between "raid" and "capture". Raiding means attacking the tile and returning home (if alive) while "capture" means your main units stay garrisoned on the tile, earning you its benefits - constant boosts to resources or culture. To successfully capture a tile, you need at least one non-air, non-artillery unit (melee, ranged, cavalry or frontline vehicle) surviving.
It is wise to make your army effective from different types of range, as they all have their strengths and weaknesses. Different ranges engage differently in rounds of combat; for example, an army of only archers or artillery is most likely going to be wiped out in melee before their second volley, and likewise, if you only include melee infantry, they will receive heavy losses from enemy arrows and shells before they attack. The same goes for cost - sometimes, cheap units in large numbers can kill an expensive one quickly.
However, another thing to consider is marching speed. This will be decided based on your slowest unit. Therefore, including artillery in a campaign to a faraway tile will severely slow you down. Marching speed can be increased with the The Steam Engine wonder.