The proper term is “form factor.” Almost all PCs made within the last 6-7 years will be ATX or micro-ATX. These two are semi-interchangeable, as you can put a micro-ATX board in a full ATX case, but generally not the other way 'round. The older style is either AT or baby AT. You can usually distinguish the two by the power connector from the PSU. The AT connector is single two-row Molex, while ATX is two single-row Molex. There is another, much more rare type called NLX, which is no longer in use.
The easiest way is going to be to bring it to a computer shop and ask them. If you’re going to be taking it apart, you could bring either the mother board or the case in and they can tell either of those. And the phrase you’re looking for is ‘form factor.’
Now that I think about it, if you google around using ‘form factor’ and some other search terms, you’ll be able to find a site that can tell you. Or, if you like, post a picture of the innards, and the back of the case and someome here can tell you.
Beware: some name-brand PCs use power supply and motherboard power connectors that are non-standard.
You may wish to have a friend who is good with electricity meter the voltages on your ‘ATX’ connector to make sure that it is in fact a standard ‘ATX’ wiring configuration.
One concern I have is the mention of a hardware encryption chip on the motherboard. You may lose access to saved data and the OS that came with the computer. I consider emachines to be the worst computers to deal with, because of all the special equipment modes they have. I would tell you to not upgrade an emachine.
I’m not that concerned about that, as the PC is currently blank as a newborn child (no OS). My intention is to purchase a new larger HDD anyways, as the 40g that it currently has onboard would hold my music and pictures, and that would pretty much be it.