The above answer is correct, but perhaps incomplete.
If you're running a node on your own machine, and the chain forks, nothing happens to your wallet or the coins it's holding. The other fork magically 'materialized' and on that chain, at the moment of the fork, you have exactly what you had (and still have) on the existing chain. If you had 100 ether before, you have 100 ether now, and 100 of the new coin as well.
In order to get that other coin, you must be running the blockchain that represents that coin. That means you would have to download and run the changes that caused the fork in the first place. This does not mean that you have to totally remove the existing chain, but you may have to for space considerations or other reasons. Once you've download the 'forked' code and synced the 'forked' chain (which has most of its data identical prior to the fork but is totally different after the fork), then you can interact with that chain--but--only if your wallet understands that chain.
When the DAO fork happened, you could download a new version of the Mist browser and there was a choice that used to pop up asking you which chain you wanted to run. There was a lot of controversy about what the default should be ETH or ETC. That option has since been removed.
So, it's not quite as simple as "You just have a new coin in the same amount as you had of the old coin," but it sort of is. You just have to figure out how to go and get those new coins.