Or... definitions that actually involve things that happen in game.
When you have any power, it has an entry used to indicate its attack-type and its range.
Attack-type is one of: Melee, Ranged, Close, Area, or Personal. This is true even for non-attack powers, this is just the technical term for it.
Range is what comes immediately after.
Okay, so how does this apply to Melee powers?
Basically, the range is a number, and ALL melee powers show up as: "Melee 1" which tells you the range of the power. Sometimes, however, it's not a constant, but a variable number. For Melee powers, there's two possibilities: Melee touch, and Melee weapon.
For "Melee touch" touch refers to your personal reach, without a weapon. Generally this will mean 1 square, however there are some paragon paths and epic destinies that can change this permanently, and there are some powers that can change this temporarily. 'Reach' is basically 'Melee range.' That's all it means.
For "Melee weapon" powers this refers to your personal reach, plus the reach of the weapon. So, if your weapon has the ability 'reach' then it will add one to your reach for this power. Where some confusion lies is thinking that this is indicating the power uses a melee weapon; this is not true. 'Weapon' is a number in this instance.
So, as for what is reach? Reach is a quantity a creature has; weapons may add to the creature's reach, and it essentially is a number that gets plugged into Melee touch and Melee weapon powers.
Also, there's no such thing as a Melee power without a range. This is a typo, as the power cannot be executed without one.