The Grand Staff begins as a theoretical 11-line staff like you see above. In practice, this number of lines would become too difficult to navigate, but if we remove the middle line, we end up with two 5-line staves (plural of staff is staves). Using two staves like this is a nice way to write music where both hands may be used to produce different pitches. For example, the piano frequently uses a pair of staves like this, with the top staff indicates notes played with the right hand, and the bottom staff indicates notes played with the left hand. The harp and the marimba also read a similar notation.
In most piano music, we place a treble clef on the top staff (for the right hand) and a bass clef on the bottom staff (for the left hand). As you can see, this places the note C where the middle line was. In contrast to the center line filling the width of the score as we saw with the eleven-line staff, the ledger line preserves the separation between the staves. This note, falling one ledger line above the bass staff, and one ledger line below the treble clef, is traditionally referred to as "middle C". This all seems logical: the right hand plays the treble staff, the left hand plays the bass staff.
But we aren't there yet...there are still a few issues:
In order to make clear which hand plays middle C, extra space is added in-between the two staves, and a curved line called a brace is also added (at the far left) to show that these two staves still belong together. Middle C is placed one ledger above the bass staff, and one ledger below the treble staff. This now means we have two versions of middle C, as you can see above. This allows the player to easily see which hand should play any notes that might fall towards the center of the grand staff. Coincidentally, this is also the area towards the center of the player's body, and therefore easy to reach with either hand.
Feeling like you need to go over that just one more time, but in a slightly different way? Check out these great explanations of the same concept: