Think Aasimar (or Tiefling if you're Paul :D ).
They have no monster hit dice, but have a level adjustment of +1. This means they can't be a level 1 Fighter until they are a level 2 character.
For other monsters, you work out monster levels that cover both monster HD and LA…
Example: Centaur (HD4 LA+2). You make six levels of "Centaur", which they progress through for their first six character levels. Their full powers are spread evenly across those six levels, and when they complete their sixth level, they are a "fully grown" centaur. All good!
However, a key feature of this is that one of the monster HD is used in the first level. Hence, a first-level centaur still has 1d8 of hit points.
A tiefling has no HD. At first level, theoretically, it spends a whole level as "tiefling", but has 0 hit dice and 0 hit points.
HD0/LA+1 creatures are not dealt with properly in the Monster Manual, Savage Species, or Races Of Destiny. Level 1 chars for these races are not mentioned, although level 1 centaurs are. The forums are awash with "solutions", which often take the form of "treat Lv1 as half-tiefling, half-character-class, and it will work out by Lv2". Very unwieldly, as you need to define a "level 1/2" for each character class (level 1/2 for wizards was mentioned in one place as cantrips-only).
Any fresh ideas?
It applies to Aasimar, Tiefling, Genasi (x4), other planetouched variants, Hobgoblins, Merfolk, SRD half-giants, catfolk, a number of fey1.
1: A grig is officially a 0HD/LA+4 creature. Savage Species gives the Grig its 1/2 d6 at level 1 of its 4-level progression. This is the HD listed in the MM (the impressive thing is you have to survive four levels with only 3 hit points!) This "breaks the rules" - a Level 5 grig wizard gets its 5d4 for wizard, plus 1/2 d6 for grig. Compare this to a Level 5 elf wizard, who gets 5d4 for wizard, but does not get its 1d8 for elf.