How new are you to modding?
If you haven't already, go and watch Gopher's youtube series on getting started with modding - http://www.youtube.com/playlist?list=PLE7DlYarj-DdhDG41roBVJfNCqvO5MmKP
This series covers pretty much everything you need to know to get started.
Watch the video series in order, don't be tempted to skip ahead to a later video in the series just because you're more interested in it - you're likely to miss some important info in an earlier video.
If you don't watch this series, don't be surprised when you break your game.
Key things to bear in mind when modding, if you ignore these YOU WILL HAVE PROBLEMS AND MAY BREAK YOUR GAME!
1) Load order is very important. Use BOSS or LOOT to make your mods load in the correct order. You'll need to use 3rd party software such as BOSS, SKSE, TES5Edit, Wrye Bash, etc. over the course of your modding - these are all completely safe to use, and are a necessary part of getting the best from modding. Don't worry about installing all of these straight away though - you'll know when you need to use them.
2) You can't uninstall a mod (with an esp file) and continue the same savegame, as data from mods gets baked into your save. If you want to remove a mod, you need to revert to a save from before you installed that mod, or start a new game.
3) Don't install a load of mods at once. Install mods one at a time, and test thoroughly (for hours, if not days) to make sure that you're happy with the mod, that your system can handle the mod in extended play, and that there aren't any bugs with the mod.
4) Always read the mod description and readme file for a mod. Common mistakes are crashes due to not reading the mod description with specific install instructions or requirements for another mod. If your game crashes on startup, you're missing a mod that another mod requires in order to work.
5) The only really essential mods are the Unofficial Patches (and arguably other bugfix mods such as Run For Your Lives and When Vampires Attack). Everything else is down to your personal taste. People will be quick to recommend mods they like, but don't rush to install anything...take your time to read up on the mods, and again - take things slowly.
If you're into simulation and immersion, the following mods I consider to be essential, and suitable for every character:
All the Unofficial Patches by Arthmoor - essential in any game.
http://www.nexusmods.com/skyrim/mods/9557/? by Arthmoor
Allows the player to skip the helgen intro/start sequence. Very useful for roleplaying - starts you off in a location of your choice with a faction of your choice - you can roleplay that you're not even the Dragonborn if you choose, but allows you to return to helgen and pick up the main quest if you want to later on.
http://skyrim.nexusmods.com/downloads/file.php?id=3562 by JustinOther
Immersion mod. Allows the player to pick up or otherwise interact with the various bits of clutter (rags, bones, urns etc.) that can be found in the game.
http://skyrim.nexusmods.com/mods/17930 by Piratelord
Makes all barrels and urns activatable, and rebalances loot found within - adds a load more clutter items (broken ancient amulets, steel rings etc.) with a small chance of decent loot. Makes more sense - for example draugr won't have gold in form of septims but as Ancient Nord Coins which are tradable for cash, and dwemer containers have dwemer coins.
http://skyrim.nexusmods.com/downloads/file.php?id=11163 by Chesko
Essential in all my games. Be wary of travelling into cold regions - you'll get cold and wet, so take a fur cloak and don't go swimming in frigid water! Completely customizable, as lite or hardcoe as you like.
http://skyrim.nexusmods.com/downloads/file.php?id=32448 by Dominoid
Chop down trees to make wood (compatible with Frostfall), pick up wood from the various vanilla firewood stockpiles (beware, some may be owned!), makes all keg barrels activatable, gain experience when using training dummies, etc. Compatible with Piratelord's Loot Adjustments.
http://www.nexusmods.com/skyrim/mods/26228 by Perseid9. A configurable needs mod, pretty complex but without the need for micromanagement. There are other alternatives out there, such as Kuertee's http://www.nexusmods.com/skyrim/mods/13246, and Isoku's mod http://www.nexusmods.com/skyrim/mods/51473. I'm going to swap RND for iNeed in my new playthrough, but can't recommend it here simply because I haven't tested it myself in-game yet - but I'm sure it's going to be a must-have.
http://skyrim.nexusmods.com/downloads/file.php?id=36334 by Ripple
Adds NPCs such as inn patrons, shoppers, street urchins, students at the college, prosttutes etc. to appropriate locations. All voiced. Inspired by Interesting NPCs, which I also highly recommend, but realise it's not suitable for every character/playthrough. Inconsequential NPCs on the other hand just makes the world feel that little bit more alive, without overpopulating.
http://www.nexusmods.com/skyrim/mods/3222 by Gopher
Lets me turn off the compass, turn off the crosshair unless I have a ranged weapon or spell equipped or hover over an activatable object, etc.
http://www.nexusmods.com/skyrim/mods/28202 by abot
No longer can you stand in a fireplace or burning embers and not take damage. Adds fire damage to fires and hot geysers.
http://www.nexusmods.com/skyrim/mods/22745 by jonwd7
Adds footprints to player, NPCs and many animals. Makes such a noticable visual change I cringe when I see videos without it. Also helps for tracking.
http://www.nexusmods.com/skyrim/mods/27563 (with the http://www.nexusmods.com/skyrim/mods/31968 add-on for Dragonborn, and the http://www.nexusmods.com/skyrim/mods/51472 add-on) and http://www.nexusmods.com/skyrim/mods/37873 - all by isoku. Also throw in http://www.nexusmods.com/skyrim/mods/50954 while you're at it.
Makes NPCs run for cover when it rains. They'll equip cloaks when it's wet or snowing. Fits in nicely with Frostfall - NPCs will also equip Frostfall items if that mod is also installed. Splash of Rain adds rain splashes to the ground - much like Footprints, it really adds to immersion. Holidays adds the holidays from the Tamrielic calendar (as seen in earlier Elder Scrolls games) with festivities, sales, events, etc. on the appropriate dates.
http://www.gamesas.com/topic/1485716-what-are-your-cant-live-without-mods/?id=1175 by Elys
Bah to the official Legendary mode. THIS is the way to allow higher levelling.
http://www.nexusmods.com/skyrim/mods/24913 by lifestorock (purist lore-friendly version edited by myself)
Adds new enemies for a lot more variety in-game. Comes with a "purist lore-friendly" MCM menu if you find some of the creatures out-of-place, and you can toggle various creatures and spawns on and off according to your own tastes.
http://www.nexusmods.com/skyrim/mods/23906 by Arthmoor
Prevents peasants and shopkeepers in towns from trying to take on a dragon and dying immediately, leaving the gameworld empty and potentially breaking quests. All non-warrior NPCs will now run for cover when a dragon attacks.
http://www.nexusmods.com/skyrim/mods/28235 by Arthmoor
Prevents peasants and shopkeepers from trying to take on a vampire attack in a town and dying immediately, leaving the gameworld empty and potentially breaking quests. All non-warrior NPCs will now run for cover when a vampire attacks.
http://www.gamesas.com/topic/1485716-what-are-your-cant-live-without-mods/?id=23390 by Eckss
Unlocks plenty of vanilla dialogue provided by Bethesda in the game files but which they decided not to incorporate. Not technically a bug, so not fixed by the Unofficial Patch - but highly recommended to make the guards more interesting (and be more respectful to you when you deserve it!)
I'm obviously biased, but I consider a couple of my own mods essential for me in every playthrough:
http://www.nexusmods.com/skyrim/mods/31463 by jackstarr
Removes the tag that says "empty" on an empty container, letting you know it's empty (through some sort of hidden psychic power) before you even look inside.
This mod also includes the optional file "No HUD Key Prompts" (by mardoxx) which removes all HUD key prompts from the game (for example, removes where it says "E" to let you know what to press to activate, etc.)
http://www.nexusmods.com/skyrim/mods/44115 by jackstarr
Removes the visual tag that lets you know if a lock is apprentice, adept, master, etc. - now you need to try and pick the lock and judge the difficulty level yourself.