You already have the "essential" mods with the Unofficial Patches. Myself, I do not think Bethesda's mods are "essential," but that's up to you. Each of us has our own personal list of "essential" mods that are completely different from everyone else's personal list of "essential" mods. But, with that said, there are a few mods that are "essential" for every game and every character I play.
http://oblivion.nexusmods.com/mods/10763/? by DarN. This mods makes Oblivion's consolized menus and HUD a tiny bit more bearable. You can move or remove HUD elements around your screen from within the game.
http://oblivion.nexusmods.com/mods/13879/? by abo or http://oblivion.nexusmods.com/mods/14065/? by tejon. Both mods level the character dynamically. No more requirement to sleep in a bed to level up. Each skill advance contributes a small portion to leveling several Attributes at once, doing away with the focus on picking certain skills purely in order to get Attribute bonuses. Health is retroactive in both mods, doing away with the need to level Endurance early. These are my favorite roleplaying mods. With these mods I can forget about number crunching and roleplay my characters.
http://oblivion.nexusmods.com/mods/41005/? by tejon. Among the many things Fundament does is allow us to fine-tune exactly how fast each skill levels up. It is great for Mages because it adds slight magic skill increases for missed shots. All three of these mods (Realistic Leveling, nGCD and Fundament) are highly configurable via .ini files.
http://oblivion.nexusmods.com/mods/25078/? by TheNiceOne. With this mod you can adjust the amount of gold a merchant has (and under what conditions he has it), adjust the amount of gold you receive from quests, adjust the frequency of enchanted items found in loot, adjust the price of houses and add property tax, disable the Haggle bar, introduce local economies (each region has its own economy), a global economy (prices all over Tamriel are affected by the stages of the Oblivion crisis) and much more. All configurable.
http://oblivion.nexusmods.com/mods/5701/? by Quazzy. This mod upgrades any level-scaled quest rewards in the game automatically. You no longer have to put off quests to get the best version of a quest reward.
http://oblivion.nexusmods.com/mods/34841/? by JRoush. This mod allows skills and attributes to continue leveling past 100.
http://oblivion.nexusmods.com/mods/43612/? by nuska. This is a graphics mod for the player character that does much to get rid of the "potato face" look of vanilla faces.
http://www.gamesas.com/topic/1439792-relz-the-unique-landscapes-project/ by the Unique Landscapes project Team. This mod utterly transform Cyrodiil and infuses it with detail and creativity. I truly is a wonder to behold. Mere screenshots do not do these mods justice.
http://oblivion.nexusmods.com/mods/16513/? by the Better Cities Team. The equivalent of Unique Landscapes, only for cities. Adds tons of detail to all the cities. Adds a second story to the Market District and makes the Arboretum actually look like an Arboretum. For the other cities you can choose between vanilla-style closed cities and open cities.
http://oblivion.nexusmods.com/mods/19628/? by XMarksTheSpot-Brumbek-Ismelda. If you've played Skyrim you've already seen Bethesda steal ideas from this mod. Adds smoke from chimneys and lit windows at night. You can choose from various window-light colors. These are small touches but they make a huge difference in atmosphere.
http://oblivion.nexusmods.com/mods/2037/? by Quarn and Dejunai. Again, if you've played Skyrim you have seen this mod incorporated into that game. Parts of plants disappear after they have been harvested.
http://oblivion.nexusmods.com/mods/23783/? by LprMan. This makes vegetation larger. With this mod the Great Forest is genuinely great. It feels like a California Redwood forest. Bushes are also increased. This is one of my favorite 'atmosphere' mods.
http://oblivion.nexusmods.com/mods/16005/? by mmmpld. This cool little mod allows you to see your character get in and out of bed. Another wonderful 'atmosphere' mod.
http://oblivion.nexusmods.com/mods/35908/? by Shezrie. Five new towns scattered across Cyrodiil. Shezrie is a mistress at house and village building. Her Pells' Gate is, in my book, the most wonderous town I have ever seen in a video game, modded or vanilla. Cyrodiil just does not feel like Cyrodiil to me without her Woodland Village.
http://oblivion.nexusmods.com/mods/28977/? by Emma. Emma has been making companion mods for the Elder Scrolls since 2002. And her Vilja, in my opinion, is the most astounding, life-like, breathtakingly real NPC companion I have ever seen in any video game. Vilja is almost a work or art. If she likes you she may let you accompany her on her adventures. If there is any companion mod better than this it is Emma's own Vilja for Skyrim.
http://oblivion.nexusmods.com/mods/29688/? by side777 and http://oblivion.nexusmods.com/mods/27235/? - by onra. These last two mods are purely for those who like to explore, for those who want to do something besides fight monsters every 30 seconds. Tamriel Hightmaps adds the entirety of Tamriel to the game. When combined with Side's ships you could embark on a Magellan-like voyage from Anvil around the coasts of High Rock, Skyrim, Morrowind, Black Marsh, Elsweyr, Valenwood, Summerset Isle and back to Anvil. Be warned though, depending on ship type and weather conditions it may take you all day (in real life). I use these two mods to sail around parts of Tamriel (or all of it) with many characters. It is wonderfully relaxing.