I am trying to fill in the "Alternative Character Sheet" (one of the official character sheets provided on Wizards of the Coast's website). It asks me for my "Maximum Hit Dice" and the "Remaining Hit Dice". What do these terms mean?
Hit Dice can be "spent" at the end of a Short Rest to recover HP. You have a Maximum number of Hit Dice equal to your total level, with the size of the dice based on your class (As Detailed in the "Hit Points" section of every class description).
Because you can expend Hit Dice to heal, you may presently have fewer Hit Dice available to spend than your total maximum. That's what the second field is for.
So, if you are a 5th level Fighter (using my preferred method of notation), you would put "5(d10)" in the Maximum Hit Dice section.
If you have not spent any Hit Dice, the same value would be put in the Remaining Hit Dice segment. As you spend Hit Dice to heal, you subtract the number expended from that section...and add back to it (up to your max hit dice) when you regain up to half of your max Hit Dice as a result of a Long Rest.
When you level up, increment Max Hit Dice by one.
For further details on how this works, see the segment on Resting in the PHB on page 186
Hit dice are a reservoir of health.
After finishing a short rest, a character can spend their hit dice to roll them. Each rolled hit die heals the character for a number of hit points equal to the number rolled plus the character's constitution modifier. One does not have to choose in advance how many dice to roll — instead, you can roll them one by one until you run out of hit dice to spend.
Each level grants the character an additional hit die to their hit die maximum. Beefier classes get larger dice than the less physical ones: Barbarians' hit die is d12, for example, while the squishier Wizards and Sorcerers have d6. The full table can be found in the table on Player's Handbook page 45. Multiclassed characters have hit dice of the types of all the classes they have: for example, a Fighter 5/Rogue 2 would have a maximum of five d10 hit dice from Fighter levels, and two d8 hit dice from Rogue levels.
Hit dice are recovered by talking a long rest. A long rest returns hit dice up to half of the player's total hit dice, rounded down as usual but a minimum of one hit die (not in my copy of PHB, but has since been covered in Errata).
The rules don't say that a character has to recover a particular type of dice or distribute the returned dice equally after a long rest. For example, a Fighter 5/Wizard 3 multiclassed character with no remaining hit dice could choose to recover four d10s, instead of two d10s and two d6s.