Time Machine Questions. (How does it work)

Take steps to acquire an external drive as soon as possible. If you buy one, a drive 2 to 3 times or larger than your boot drive is preferable for long term trouble-free operation. Do not pay extra for a drive that is fast.  (You can get by for a while with a "found" smaller drive if necessary, but it will eventually become annoying).

Attach your external drive and use

System preferences > Time machine ...

... to turn on Time Machine and specify what drive to store your Backups on.  It may ask to initialize the new drive, and that is as expected. APFS format is default format if running MacOS 11 Big Sur or later.

Time machine works quietly and automatically in the background, without interrupting your regular work, and only saves the incremental changes (after the first full backup). Time machine backs up every connected drive that is in a Mac compatible format. it can not back up Windows format drives.

Time Machine's "claim to fame" is that it is the backup that gets done. It does not ruin performance of the rest of the computer while doing its backup operations. You do not have to set aside a "Special Time" when you only do backups. When you need it, your Time machine Backup is much more likely to be there.

How to use Time Machine to Backup or Restore your Mac:

Back up your Mac with Time Machine - Apple Support

<<. When I turn Time Machine on, and it saves my Mac, is that stored on my Mac until I decide to connect my external HD to it? >>

Time machine makes COPIES of your files. The original files are not changed in any way.

But time machine is not an Archiver. As time goes on, files deleted from your Mac will eventually 'age out' and be removed from the backup set as well.

<< I then add more photos to my Mac, the next time I save to my external HD, does Time Machine overwrite everything from the first (the original backup) or does it save as a completely different save (not sure how to put that into words). So you end up with different dates you can go back to. >>

it does something completely different. Time machine creates a complex DATABASE of your saved files on the backups drive. This database consists of the basic Full backup, and that can be adjusted by the additional incremental backups you have created since then.

But wait, There's MORE! Time machine will re-create the state of any Folder (or your entire Mac) as of any date whose components are still saved in its database. It presents the most recent view in a window, with older views stacked behind it:

...

Hi,

Wikipedia would answers your questions about Apple Time Machine.

https://en.wikipedia.org/wiki/Time_Machine_(macOS)