How to Cut and Move a Folder on a Mac (Without Losing Files)

Quick answer: Finder cannot cut a folder, but it can move one: select the folder, press ⌘C, open the destination and press Option+⌘V. The folder and everything in it moves. If a folder with the same name exists, Finder asks what to do; replacing deletes what was there, so choose carefully.
Steps
Click the folder once to select it.
Press ⌘C.
Open the destination folder.
Press Option+⌘V. The original is removed and the folder appears here.
If you press ⌘V by itself, the folder is copied and the original stays.
When a folder with the same name exists
| Finder asks | What it does | When to use it |
|---|---|---|
| Stop | Cancels the move | You are unsure |
| Replace | Deletes the destination folder and puts yours there | You want your version to win |
| Keep Both | Adds a copy with a different name | You want to compare |
| Merge (Option held) | Combines the contents | You want to join the two folders |
To combine the contents of two folders rather than replace one, hold Option while dragging a folder onto one with the same name. Finder then offers Merge. Check the result before deleting the source.
Moving a folder to another disk
Moving across disks copies the folder first, then deletes the original once the copy is complete. For large folders this takes time, and the destination needs enough free space. See Move vs Copy on a Mac: What Happens With External Drives, USB Sticks and Network Shares.
Folders you should not move
Do not move system folders, or folders that belong to an app, such as an app's folder in Library, unless the app tells you to. Moving a folder an app expects in a fixed place can break it.
A real Cmd+X for folders
FileCutPaste lets you press ⌘X on a folder and ⌘V in the destination, with Finder doing the move.
FAQs
How do I cut a folder on a Mac?
Copy it, then Option+⌘V in the destination.
Does moving a folder delete its contents?
No. Everything inside moves with it.
How do I merge two folders on a Mac?
Hold Option while dragging one onto the other, then choose Merge.