For folks that just installed Steam, check out this https://support.steampowered.com/kb_article.php?ref=5357-FSQM-0382 that explains installing from the disc:

What is Steam and SteamWorks?
Steam is a Digital Distribution (DD) and Digital Rights Management (DRM) platform used to distribute, update and play games online. Its main purpose is to allow players to buy games online and download them directly to their PCs with the Steam client, but it's also used by games distributed on physical media, such as DVD discs. Games that use Steamworks, which includes Steam-activated physical DVD discs, are activated online using the Steam client, and can't be launched without the Steam client running alongside the game. Thus the Steam client is required for both physical DVD and digital installations; it will be installed automatically for DVD installations from the disc, and there is no difference in the resulting install whether Skyrim is purchased online through Steam or on physical DVD. See http://en.wikipedia.org/wiki/Steam_(software) for more information.

Do I need Steam to run Skyrim on PC?
Yes. Skyrim is strictly tied to Steam for both physical retail DVD discs and online digital-only purchases. To play Skyrim, you launch the Steam client and then launch Skyrim from the Steam library, leaving the Steam client running in the background.

Can I just run Skyrim without Steam running in the background?
No. The Steam client must be running to play Skyrim, and discussion on circumventing Steam by running Skyrim by any other means will not be allowed within these forums.

Do I need to be connected to the internet to activate Skyrim?
Yes. There is no such thing as offline activation.

Do I need to be connected to the internet to play Skyrim?
No. Steam is able to run in offline mode, meaning that you need to get online only once to activate Skyrim and install any required updates/patches, and don't need to be connected to the internet to play it. See https://support.steampowered.com/kb_article.php?ref=3160-AGCB-2555 for the official guide, and the last entry in this FAQ for more detailed instructions on how to set the Steam client to offline mode so that it can stay offline for a long time. The Steam client is often updated, and the offline mode is known to be still a little buggy nowadays, hence it is advisable follow the latter instructions to set the client in permanent offline mode once you have activated Skyrim if you need extended offline time, and regardless of any further Steam client updates.

I have preordered my physical DVD copy of Skyrim; how am I supposed to install it? Do I need to install the Steam client first?
No. What you're actually installing when you completely install Skyrim from the DVD is the Steam client itself (as long as you haven't installed Steam in your system yet), which will then automatically copy the required game files from the DVD to your hard drive. You'll be asked to enter your Steam account name and password to log onto Steam authentication servers, then setup will start. If you don't have a Steam account you'll be asked to register one before installing Skyrim. Note that some of the files may be pulled from the internet through the Steam client to complete the installation, rather than all being on the DVD.

I want to order the Collector's Edition of Skyrim, but I don't have an optical/DVD drive. Will I still be able to install it when it arrives?
Yes, as long as you have a broadband internet connection. You can activate the DVD key of any SteamWorks game, including Skyrim, in your Steam client and download and install it over the internet without ever needing the DVD at all.

I have purchased a copy of Skyrim through Direct2Drive, GamersGate or some other digital download portal, not a physical DVD version. Won't that be different from the Steam version? Will they be incompatible for modding?
No. There will be no difference nor incompatibility. Skyrim is exclusively on the Steam platform. These sites are selling Steam activation codes for Skyrim, similar to the SteamWorks DVD key. When you buy a code, you will activate it in your Steam client, which will then download and install Skyrim, just as if it was bought through Steam. So, there will be no difference from a version bought directly through Steam itself.

Where am I supposed to install Skyrim?
Any Steam game is always installed into the main Steam folder without prompting for install location. However, you may choose where to install the Steam client itself, which will determine the install location.

What if an update/patch comes out? May I just download the patch and install it, with the Steam client still set in offline mode?
No. There will be no manual updates or patches available for download anywhere; the only way to update Skyrim is to get online and let the Steam client download the updates for you. Then you will able to get back offline and play Skyrim as usual.

I'm on a slow/dial-up internet connection. Can I still activate Skyrim when I install from physical DVD disc? Will I be able to download required updates/patches?
Possibly. You will be able to activate Skyrim on a slow or dial-up connection. Steam activation data files are usually in the order of a few megabytes, so should only take a few minutes to download. However, at activation time it is required to install any required updates/patches.

Will I be able to play Skyrim as soon as it releases?
Yes. There will be more details on preloading Skyrim via Steam closer to release.

I have a question or concern about Steam, or DRM in general, that's not answered here.
Then you're encouraged to refer to the Unofficial Steam/DRM Discussion found in the http://www.gamesas.com/index.php?/forum/18-community-discussion/ forum or visit the http://store.steampowered.com/forums/.

Thanks to dAB for helping put together this FAQ

If you are seeing the error, "This game is currently unavailable. Please try again at another time," there is a simple way to fix it.

1. Close Steam. Right click on its icon in the System Tray, and Exit.
2. Go to Steam's folder. C:\Program Files (x86)\Steam
3. Find the file ClientRegistry.blob and rename it or delete it.
4. Start Steam again. It will re-download the client.
5. You're fixed. Have fun.

If you're having trouble running Skyrim after patching the game, please try the following:

1. Right click on the game in the Steam library and select Properties
2. Navigate to the "Local Files" tab
3. Click "Verify integrity of game cache"

Also note that verifying your game cache will reset your SkyrimPrefs.ini file. If you are manually editing settings in that file, create a backup prior to verifying the cache.

Earlier Versions of the Game had some issues related to sound settings on the sound card. This should not be a problem with 1.3 and later versions. However, if you are having issues, here is something to try

If you're crashing to the desktop, try this workaround
1. Go to Start -> Control Panel -> Sound 2. In the Playback tab, right click on Speakers (or whatever has a green tick next to it), then click Properties 3. In the advanced tab on this new window, select 24 bit, 44100Hz (Studio Quality) in the drop down menu

If you're having sound issues, such as static during NPC dialogue
1. Go to Start -> Control Panel -> Sound 2. In the Playback tab, right click on Speakers (or whatever has a green tick next to it), then click Properties 3. In the advanced tab on this new window, select 24 bit, 44100Hz (Studio Quality) in the drop down menu