They were drawn to Earth after hearing of the Xen invasion from Half-Life, and then managed to conquer the planet within seven hours.
The Combine rely on technology which allows them to transform other creatures into their mindless servants and modify their bodies to turn them into something different entirely.
Most of the Combine enemies Gordon Freeman fights are masked transhuman soldiers. These biologically modified Combine units are known as "Synths", and the sentient creatures that control the Combine Empire are the "Advisors", which resemble giant featureless slugs/worms but have powerful telekinetic abilities and a deadly tongue that can drill out a person's brains.
Another notable Combine unit is the "Strider", an extremely dangerous tripod creature equipped with a powerful machine gun and invulnerable to most weapons.