Given this is a travel forum, the airline terminology is the best one.
A layover refers to a break between two flights taking you to your destination. Normally this would be short (a few hours), but the definition will vary depending on the airline and the route.
A stopover is where you actually break your your journey at a point that isn't the destination for longer than the time allowed for a layover. Stopovers almost always have additional fees associated with them - not only from the airline, but also as you may end up paying additional taxes/airport/customs fees as a result of the longer stop.
How long a layover can be varies. Some US airlines for domestic legs will define it as less than 4 hours, OR the next available flight to you destination. ie, if the next available flight to your destination isn't until 6 hours after you arrive, that would still be a valid layover - even through it is more than 4 hours. Many airlines will also allow a layover to extend overnight, again as long as that's the next available flight out.
For International flights most airlines allow up to 24 hours count as a layover, and many countries also use 24 hours as the limit (eg, if you are in Singapore for less than 24 hours then you do not pay the taxes you would for a longer stop).
Personally I've used the international 24 hour rule countless times to be able to spend a day (and sometimes even a night!) in a country without having to pay for the stopover!
Edit: IATA () has a "Glossary of Terms" on their website. It's doesn't have a definition for Layover, but the definition for Stopover is :
Equivalent to a "break of journey", means a deliberate interruption of a journey by the passenger agreed to in advance by the airline, at a point between the place of departure and the place of destination.
One important point I forgot to mention earlier is that many airlines will treat checked bags differently for a stopover v's a layover/connection. By default, your bags will be checked to either your destination or your stopover point if there is one. Some airlines will allow you to "short-check" your bags to a layover/connection point, but many will not.