1. Boot into recovery mode by holding down Command+R at startup; you'll eventually see a Mac OS X Utilities window. (If this doesn't work, restart and hold down Command+Option+R, which should force your Mac into Lion Internet Recovery.)
2. Connect a drive--a hard drive, a thumb drive, or the like--with at least 5GB (preferably 8GB) of free space. The drive must be formatted as Mac OS Extended (Journaled), and must have a GUID Partition Table. Follow Steps 1 through 4 in this slideshow to properly format the drive.
3. In the Mac OS X Utilities window, click Reinstall Mac OS X and click Continue.
4. On the Mac OS X Lion screen, click Continue. You'll see the message, "To download and restore Mac OS X, your computer's eligibility will be verified with Apple." Click Continue, then click Agree (twice) on the next screen to agree to the Lion license agreement.
5. Select the drive onto which you want to install Lion. The important thing here is to select your external drive.
6. Click Install to begin the download. Depending on your Internet connection, the download can take anywhere from under an hour to several hours (or even, if you're unfortunate enough to be on a very slow connection, considerably longer).
7. IMPORTANT: Monitor the download's progress. As the progress bar gets near the end, get ready, because once the status reads About 0 seconds remaining, the progress bar will disappear, the installer will spend a minute or two cleaning up, and then your Mac will restart. As soon as the screen goes dark, unplug your external drive. If you wait too long, your Mac will boot into the Mac OS X installer on that drive, starting the installation process. Interrupting that process can leave your Mac unable to install OS X unless you restart it and--I'm not joking--zap PRAM.
8. Once your Mac has booted from its normal startup volume, reconnect the external drive. Alternatively, you can connect the drive to another Mac and proceed with the following steps using that Mac.
9. Open the external drive, and you'll find a folder called Mac OS X Install Data. The important file is the one called InstallESD.dmg, just under 4GB in size, which is a bootable disk image containing the Lion installer. (If the Mac OS X Install Data folder has a "no access" icon, select the external drive in the Finder, choose File -> Get Info, and expand the Sharing & Permissions folder in the Info window. Click the padlock icon at the bottom of the window, provide an admin-level username and password, and then uncheck the box next to Ignore Ownership On This Volume.)