I changed the chain on my bicycle before doing any proper reading on it, and I did not check whether the chain was the correct length. It seems to be working ok... but what could potentially go wrong?
If the chain is too long then in some gear combinations (e.g. small/small) there will not be enough capacity in the dérailleur to pick up all the slack, so the chain will be loose. At an extreme case, the chain could sag and come into contact with the chainstay.
If the chain is too short then in some gear combinations (e.g. big/big) there will not be enough chain to span both gears and go through the dérailleur. In this case it might not be possible to change to this gear, or doing so could damage the rear mech/mech hanger.
In a bike with rear suspension, there also needs to be sufficient extra chain to compensate for chain growth when the suspension compresses. Running too short a chain could mean shifting to big/big is possible, but compressing the suspension when in this gear damages the rear mech/mech hanger.
Typically there will be a range of chain lengths in which the gears will function correctly and if the chain is only a little too short/long problems would only become apparent in poor combinations such as big/big and small/small
I assume you are talking about a bicycle with a front/rear derailleur. Typical hybrid bike accepts very wide range of chain lengths.
There is a simple way to check if chain is either too long or too short.
1) Put the chain to smallest sprocket and smallest chainring. If chain rubs against the rear derailleur it is too long. There should be atleast one inch clearance between the upper pulley and chain. In this case take out a few chain links.
2) Put the chain to largest sprocket and largest chainring. Or if you can't. The chain is too short. The chain simply can't reach the largest rings.
Both problems can be avoided by not using gears that are unsuable.
If you have a road bicycle with short cage rear derailleur and a large rear cassette (for example 11-30) then the high/high and small/small combinations might not be available at all. Options are to get longer cage rear derailleur, narrower chainrings or narrower cassette.