So, I bought the book, read it, but I have a question, and just to be sure:
The max level from an skill in rank can't be more than your level, so, a starting character can only have a skill on level 1 rank, right?
Later in the book, it says that the max rank on a skill can't be more than your hit dice, so, for example, a bard has a d8 hit die, so, does that means he can only rank his skill up to 8, once he leveled up once?
As you state correctly, a character can have a maximum number of skill ranks in any skill equal to its Hit Dice.
Hit Die, singular, (i.e. the bards d8) is a basic component of a class and only determines the amount of hit points a character gets upon taking a level in the class1. It is (hopefully) never abbreviated HD.
Hit Dice (or HD), plural, means the number of hit dice you have, not the size of the die itself. For regular characters (e.g. standard races), this is equal to the total amount of class levels he has, i.e. his character level.
Monsters also have Racial Hit Dice, which are added to all class levels they possess (if any) to determine their Hit Dice2.
In statblock notation (like in the Bestiaries), it is noted as
Xis the Hit Dice,
Ythe Hit Die, and
Call constants (CON-Bonus, Toughness, etc.). If there are more than one type of Hit Die, there are more terms like the first one. The total Hit Dice of the creature then is the sum of all
Some quick Examples:
1 The hit die also usually determines the BAB progression of the class. This holds true for all Paizo classes as of this moment, but not all 3PP classes follow this trend.
2 For a more thourough explanation, see Jack Lesnie's Answer