Does level adjustment simply mean the character must be the adjustment's number of levels lower than the rest of the party, or does it act as levels in the relevant template and include the relevant number of hit dice and/or skill points?
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Level Adjustment does not convey any hit dice alone, it simply adds to the effective level of a character.
Effectively, your first option is correct in that this character will be 2 class levels behind other party members.
For example a level 1 Drow Rogue would only have a single hit die and only the class abilities of a level 1 rogue, but would count as a level 3 character and thus would need a total of 6000 xp (the amount to reach level 4) to become a level 2 rogue.
Note that some "monsterous" races may convey racial hit dice as well as possess a level adjustment. For instance a Gnoll provides 2 Racial humanoid hit dice as well as a +1 LA. Thus a level 1 Gnoll Rogue would have the abilities of a level 1 rogue, 1 rogue hit die, 2 humanoid hit dice, and the +1 LA. This character would count as level 4, but having 3 hit dice would have 2 feats (one for level 1, one for level 3).
Level Adjustment means you “count as” being that much higher than your actual HD. The LA gives you absolutely nothing: no HP, no base saves, no skill points, not even a boost to your maximum skill ranks, but you nonetheless are that higher level for the purposes of XP (how much you get and how much you need to level up) and of expected wealth gain.
As other posters mention, you add the racial hit die to the class level and the level adjustment to get effective character level (ECL), which doesn't confer anything. ECL just has the game effect of determining your xp awards and the experience points needed to advance, and is used when your DM determines average party level for encounter level and expected wealth. Your DM might also want everyone to be the same level or within some range, so that makes playing a race with level adjustments not playable at first level.
It is important to note; level adjustment is not considered for feat acquisition or ability score increases; these are calculated based on total hit dice. This includes class levels and racial hit dice, but not level adjustment. Thus when you level, you would use your ECL to determine experience needed to level, but you would gain feats and ability score increases by total hit dice.
Savage species presented a neat way to "advance" creatures with more than one racial hit die from level one, but for the most part I think it makes the characters underpowered.
If your DM doesn't mind the imbalance, you will eventually average out to the party's level, especially if using the standard XP rules, but even with the UA's level independent awards, you won't be far ahead as it still requires much more xp to level as you get higher.
However, where I think the calculation is flawed; when adjusting monster CR, you don't count racial hit die as one for one. In fact, most count as 4 per 1. If you are considering playing a character with a monster race that has racial hit dice, you might talk with your DM about what that amounts to.
For example, a 1st level rogue satyr has an ECL of 8 (1 for rogue, 2 for LA, 5 for racial hit dice), but the CR for that same creature if you add a level of rogue is only 5 (CR 4 +1 for the PC class level) or 3 if you drop the pipes! Somehow that math seems a bit off to me :)
I would suggest a good house rule for making a character with racial hit dice would be to count the racial hit dice in a similar manner; that is, level adjustment plus PC class levels plus the same for racial hit dice as the Table: Improved Monster CR Increase adds to CR.
If you counted the racial hit dice as +1 per 4 HD added (per Fey), the rogue satyr's ECL would be 5 (1 for 1st level rogue, 2 for LA and 2 for 5 Fey racial HD)*. That sounds much more accurate to me as a DM, at least as far as how effective this character would be compared to other players of the same character level without level adjustments. If you made this change, I would suggest using ECL for feat acquisition and ability score increases as well (since ECL may be lower than total hit dice).