This is just the simplest way of flashing nightly's (IMO) without Rom Manager (excluding Google Apps).

1. Download the AIO Captivate Toolbox. http://forum.xda-developers.com/showthread.php?t=944418  

2. Make sure you have the correct drivers installed.

3. Click Flash to Stock in the program and Odin One Click.

4. After that's completed follow the directions in the program and Master Clear (Optional).

5. If you don't have 3 Button Fix enabled its strongly recommended that you do that now (Optional).

6. Root your phone using the program.

7. Download the latest nightly from the link below and then click Flash Custom Rom in the program (follow all directions). 

8. You now have the latest nightly installed.

WHAT I POSTED ABOVE IS MY OWN METHOD. I AM NOT A EXPERT. IF YOU ARE MORE COMFORTABLE WITH THE OFFICIAL DIRECTIONS THEY ARE BELOW. ALSO WHAT I WROTE ABOVE IS NOT FOR UPDATING, IT IS FOR FIRST TIME INSTALLS. IF YOU WANT GOOGLE APPS .... INSTALL ROM MANAGER AND DOWNLOAD THEN FROM THERE. ROM MANAGER IS ALSO THE SURE WAY OF GETTING INTO RECOVERY, THE POWER MENU DOESN'T ALWAYS WORK.

INSTRUCTIONS:

Latest version: http://download.cyanogenmod.com/?device=captivatemtd
Check the md5sum if you want to be sure that the download worked (different OSes have different programs that do md5, on ubuntu it is md5sum that you run against the zip you download and compare to the md5sum number shown on the download website). ROMManager does this automatically for ROMs you download via ROMManager.

- First time flashing CM 7 to your Captivate (or coming from another ROM)?

IF YOU DON'T WANNA PAY $7.00, FOLLOW MY DIRECTIONS OR FOLLOW THE "HARDER WAY".  
Easy way:

1. Start with a rooted Eclair/Froyo running Eclair/Froyo Bootloaders (do not ask us how to do this).
2. Install ROMManager Premium. 
3. Flash ClockworkMod Recovery. (MAKE SURE!!! YOU HAVE CLOCKWORKMOD RECOVERY 2.X INSTALLED BEFORE YOU PICK CAPTIVATE (MTD))....Pick Captivate (MTD) (ignore the warning about not having an official, pick ClockworkMod 2.x).
4. Make sure you can reboot into ClockworkMod recovery from ROMManager. Reboot and get back into ROMManager Premium.
5. Choose download ROMs.
6. Pick the CyanogenMod Nightlies.
7. Check the Google Apps addon.
8. Check Backup current ROM!
9. Check wipe data, cache, dalvik cache.
10. Let ROMManager finish the installation.

Harder way:

1. Make sure you're running a Eclair/Froyo Firmware (2.1 or 2.2) and Eclair/Froyo Bootloaders!
2. Root your device and install ClockworkMod Recovery.
3. Do a Nandroid backup! (Optional)
4. WIPE (wipe data/factory reset + wipe cache partition)
5. Install the ROM from internal sdcard using ClockworkMod Recovery (download from the above link)
6. Optionally install the Google Addon (through ROMManager is easiest)

- Upgrading from earlier CM7?

ROM Manager method:

1. Install ROMManager Premium.
2. Flash ClockworkMod Recovery. Pick Captivate (MTD) (ignore the warning about not having an official, pick ClockworkMod 3.x).
3. Choose Download ROMs.
4. Pick the CyanogenMod Nightlies.
5. Check the Google Apps addon.
6. Check Backup current ROM!
7. Check wipe cache, dalvik cache.
8. Let ROMManager finish the installation.

Clockwork Mod method:

1. Download and push the ROM zip file to the sdcard.
2. Reboot into recovery.
3. Do a Nandroid Backup! (Optional)
4. Install the ROM zip from sdcard (your Google apps will be backed up automatically)

There are no Google Apps bundled with CM ROMs, because Google asked Cyanogen to remove copyrighted apps. After flashing the rom, don't forget to flash Google Addon package if you want it. ROM Manager will allow you to flash it if you just go into it and after flashing the ClockworkMod recovery for MTD devices, choose Download ROMs, and pick Google Apps.