Q: 2007 iMac is running incredibly (and unacceptably) slow.

1 GB RAM? IIRC your machne can run 16 GB. Get more. in the meantime, see:

Mac Maintenance Quick Assist,
Mac OS X speed FAQ,
Speeding up Macs,
How to Speed up Macs,
Macintosh OS X Routine Maintenance,
Mac troubleshooting: What to do when your computer is too slow,
Essential Mac Maintenance: Get set up,
Essential Mac Maintenance: Rev up your routines,
Maintaining OS X, 
Five Mac maintenance myths and
Myths of required versus not required maintenance for Mac OS X for information.

16 GBs of RAM?

No 24 inch model iMac can take 16 GBs of RAM.

The OP's 24 inch model iMac can take a total of 6 GBs of RAM.

Correct and reliable Mac RAM can be purchased from online Mac RAM sources Crucial memory or OWC (macsales.com).

How much free space do you have on your iMac's hard drive?

OT

Then, I remembered incorrectly, recalling what my 24" iMac could take.

A 2007 iMac will max out at 6GB of RAM however even that would be huge upgrade over the 1GB the OP has installed. To see if the OP needs more RAM  (almost a dead certainty) they should click and use the advice in Activity Monitor to read System Memory and determine how much RAM is being used. They also may have more serious issues such as a failing HD too. For that I'd strongly recommend running Apple Hardware Test in extended mode at least 3x back to back. If errors appear the next step is a trip to the local Apple Store or AASP for a hardware repair.

RAM and storage are 2 seperate things. To check the storage (otherwise know as hard disk space) open Finder - Right click your HD on the side bar and choose More Info and and you will see the capacity and available space.

To check RAM amount use the tool I suggested in my first about using Activity Monitor. 

Select the drive on the desktop and type Command+i to bring up the infor window for the drive.  It will tell you how much free space is on the drive:

It if really is just 56 MB then your are in real trouble.  You should immediately get an external HD and move large items, i.e. iPhoto Libraries and iTunes libraries, to it to free up space.  You need to maintain about 10 GB of free space for optimal system and application performance.

15-20 GB of free hard drive space would be a more comfortable amount to regain, no?

Yes. I should have stated a minimum of 10GB which works fine unless one is using iDVD which ***** up lots of  space with temporary files during encoding or working with movie editors. Then 20 GB is a better minimum.  

Yes, a big difference.  Space is not the problem. I would definitely investigate adding additional RAM.  I've run with 4 GB for a long time with no problems.  However, with my last iMac I upped it to 8 GB and it really makes a difference. Go to Crucial.com and it will automatically check your Mac and tell you how much RAM it can use and what it will cost for different amounts.