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?
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).