The G-man is a sentient, psychic Xen crystal, the middle aged man with the briefcase is just a psychic projection. He caused the resonance cascade in Half-Life 1 to try and get rid of the Nihilanth, because he was abusing the crystals by using them in his teleportation technology.

Either the Nihilanth would leave Xen after conquering Earth, or Earth would win and kill the Nihilanth. In both cases, the crystals are no longer being messed with.

The G-man's physical body is actually the crystal Gordon used in Black Mesa, that cause the resonance cascade. After the incident, the crystal has mysteriously vanished. It was unusually pure, remember? Hence his power compared to other crystals and the reason he's the one who is actually doing things to effect the story.

The crystals are used in teleportation technology, so they obviously have power over space and time, just like the G-man. The G-man works through other people, because he has no useful physical form. We destroyed 3 crystals in our fight vs the Nihilanth, so if they did have a way to work directly, they would be pretty much unable to fight a retaliation strike, so everything he does must be done without raising suspicion.

Then the Combine invaded Earth, something the G-man had not planned, and the Xen crystals were once again under threat. The Combine take all the resources they can and would be a worse abuser of the crystals the the Nihilanth had ever been.

So the G-man helps Gordon by putting him in a position where he can help bring about the defeat of the Combine. At the end of Half-Life 2, Gordon had completed his mission in sparking the revolution, so the G-man was ready to take him somewhere/somewhen else where he would be next be most effective.

However, the Vorts knew Gordon could still be of use on Earth, so they stopped him from taking Gordon. When the G-man saw how much Gordon could effect the Earth for the better his was like "Fine, things seem to be working out for the better, I'll let him tidy things up of Earth for a while longer."

Him showing up in episode 2 was just a reminder that the G-man was still there, still powerful, and that the only reason he doesn't take you away now is because he doesn't want to at the moment.