On the piano
The keyboard shows the same seven notes as a shape you can see. The coloured keys are the C major scale, and you can click any key to hear it.
The steps, on one string
C major on one string: C D E F G A B C.
Counting up from the root, the steps are whole, whole, half, whole, whole, whole, half. A whole step is two frets and a half step is one. This pattern of steps defines the scale, and the letter names are simply the notes it lands on.
On the staff
On a treble staff, the C major scale is written C D E F G A B.
This is one octave of the C major scale, written on the treble clef. Guitar and bass music is written an octave higher than it sounds, and the small 8 under the clef is the sign for that.
The major scale is a pattern of distances rather than a list of letters: whole, whole, half, whole, whole, whole, half. A whole step is two semitones and a half step is one. Start that pattern on any note and you have the major scale of that key. The letter names are simply the notes the pattern lands on.
On one string the scale is a run of frets you can see, moving up by two, two, one, two, two, two and one. Every scale shape a ukulele player uses is that run folded across four strings. The folding is why the shape stops looking like a pattern and starts to look like something you have to memorise.
C major has no sharps or flats, which is one reason so much ukulele music is written in it. All four open strings, G, C, E and A, are notes of the scale, so they suit it well.