With each expansion, in the World of Warcraft universe, the level cap of your character increases by 5-10 levels. There is a general "theme", and throughout the release and consequent patches, there are some significant events that take place in terms of storyline.
Each expansion is catered predominantly for players within its respective level group. Burning Crusade content caters to levels 60-70, Wrath of the Lich King caters to 70-80 and so on.
Each consequent expansion can, in theory, both limit and improve your ability to experience the story line of its predecessors.
Most end-game content is designed for groups of players, in anything from 5 to 40 man dungeons. There is a reliance on being able to play alongside other players, in order to have a chance against completing some of these events.
With most players leaning towards current content, there can be fewer players available to participate in older content, however, due to the level increases with each expansion, you can often find older content a lot easier to handle given fewer players.
As for if you miss out on any content, no, not really. While there is a lot of content, to begin with, you still have access to most of the major events surrounding the World of Warcraft universe. For example, you can still easily handle the major end-game raid released for vanilla World of Warcraft, Ahn'Qiraj, with only a handful of level 70s. However, the event that precipitated its opening no longer features, and can not be experienced.
You will still be able to experience the majority of each expansions story. The only difference is, you can experience it in a different order, given the ability to freely roam within your characters level restrictions.
Eastern Kingdoms and Kalimdor, the two major regions from Vanilla WoW, have undergone a massive redesign with the release of Cataclysm. You will likely experience a difference, here, in comparison to the original WoW. Most of this has been Blizzards way of bringing the vanilla content up to the standards reached with Burning Crusade and Wrath of the Lich King, and you will still experience all the "original lore" storyline in your initial character progression.
For reference, here is a list of expansions, along with their general themes and included "racial backstory". You can generally focus on the content from a particular expansion, to experience the story surrounding its theme.
The Burning Crusade (60-70): Explores Outland, the old homeland of the Orcs and Draenei, featured in the events of Warcraft 3: The Frozen Throne. Plot involves the storming of The Black Temple, and the overthrow of Illidan, the threat of the Zul'jin and his Zul'Aman empire, and the betrayal of Keal'thas Sunstrider, in attempting to summon the demon lord Kil'jaeden through the Blood Elves Sunwell. The Draenei and the Blood Elves are also introduced.
Wrath of the Lich King (70-80): Explores Northrend, a forgotten contintent to the north, featured in the events of Warcraft 3. Plot involves the discovery of an old titan complex, an old prison built to hold an Old God, and the war efforts in fighting the undead scourge, leading up to the penultimate battle against The Lich King, himself. This expansion also introduces the Death Knight, a special hero class with their own backstory, and play style.
Cataclysm (80-85): Introduces a massive apocalyptic event known as The Cataclysm, during which a great majority of The Eastern Kingdoms and Kalimdor undergoes a considerable change, including the appearance of the Maelstrom region. Plot involves the alliance and uprising of the Zandalari trolls, the defence against the firelord, Ragnaros, and the fight against the rouge dragon, Deathwing the Destroyer. This Goblins and Worgen are also introduced as more prominent races.
Mists of Pandaria (85-90): Introduces the newly discovered islands of Pandaria. Plot focuses on the renewed fight between the Horde and the Alliance, as well as the civil disputes within the factions. Further plot focuses on the return of the Mogu, the and siege of Orgrimmar. The Pandaran race is also introduced as the games first natural race.
Warlords of Draenor (90-100): Introduces an alternate timeline, as well as the planet Draenor, which eventually became Outland after being torn apart. Plot focuses on an altered timeline, where the orcs do not drink the blood of the demon lord Mannoroth. This leads to an unnatural uprising of the Iron Horde, and a major invasion of Azeroth.