What happens if an Ice Age or Interglacial Period Cycle starts? I don't know what exactly what happened then.
I'm guessing this has been answered before, but in a nutshell, the primary mechanism over the last 3 million years or so is nothing more than colder summers in the Northern Hemisphere. Now, one cold summer won't do it, you need hundreds, which usually requires an orbital variation or perhaps, ocean current change or perhaps an extended solar minimum, but the effect needs to last for at least for a few to several centuries, or something radical like a dinosaur killing meteor or a once in a million years volcano, but that's more of an extreme event and not relevant to your question.
There's places in the far north that are snow covered year round. Some of the time it gets warm enough for snow and ice to melt, but where it doesn't stay warm enough for all the ice to melt, you get ice cover 365 days a year.
When you get an extended period of colder summers, then the ice cover can move south, covering more land in the Northern Hemisphere. (The South pole isn't as important because the land near the south pole is already ice covered and that pretty much never changes). As the ice moves south and covers more land, a few things happen, often referred to as feedback mechanisms.
1) more sunlight reflects off ice, less heat stays on the Earth to warm the atmosphere, this has a global cooling effect.
2) Oceans begin to fall as ice builds up on land. Oceans absorb the most heat from the sun per square meter out of any surface on earth. Ocean heat absorption also evaporates water vapor into the atmosphere, and water vapor acts as a greenhouse gas. As oceans fall, they also cover less area and while the size change is a relatively small percentage, that also works to cool the earth.
3) as the ice grows higher, this has a cooling effect on high altitude air, which blows off the ice and cools surrounding areas and tends to keep the ice frozen longer.
4) as the oceans grow cooler, they absorb CO2 from the air, this lowers the amount of greenhouse gas in the air which has a cooling effect.
The cooler northern summers are thought to be driven by orbital changes such as low orbital eccentricity.
Interestingly, the effects such as orbital variations, milder summers and even CO2 are by themselves, quite small, but when taken together with the feedback mechanisms in place, the temperature change of the planet between ice age and inter-glacial period is quite large. As much as 10 degrees c globally, corresponding with mile high glaciers covering most of Canada, half the United States, half of Russia and Europe, as well as 300 foot sea level drops and ecosystem changes around the globe. Not all bad. Some deserts will arguably get improved ecosystems, but the change is felt globally.
The inter-ice age period is the same thing just in reverse. Orbital changes can lead to warmer summers which melts ice, and beings to reverse the 4 feedback mechanisms mentioned above. Melting ice isn't all harmless though. You get enormous lake sized pockets of water trapped in ice, waiting to give way, and floods like we've never seen.