The standard grips this chord is usually shown with.
C#maj9 is C#, E#, G#, B# and D#, a major ninth. The shape this page opens on is written x41111, lowest string first.
Note by note. C# is the root, E# is the major third, G# is the fifth, B# is the major seventh and D# is the ninth.

Also written C#M9, C#Δ9, C#ma9 or C# major 9. The same chord either way; which spelling you meet depends on who wrote the chart.
There are 11 common shapes for C#maj9 on the guitar, and 36 derived voicings in 4 families.
One note away. Change a single note of C#maj9 and you have C#maj7 (with one note taken away), C#maj11 (with one note added), C#maj13 (with one note added), C#mMaj9 (with one note moved), C#add9 (with one note taken away) or C#69 (with one note moved).
In which keys. Every note of C#maj9 belongs to C# major (the I), F minor (the ♭VI), Ab major (the IV) and Bb minor (the ♭III).
The common shapes, written lowest string first: x41111, x4354x, 44354x, x66668, xx6868, 966668, 9-8-10-8-9-8, 9-11-10-10-9-11, x-x-10-10-9-11, x-x-11-10-13-11, 13-11-11-13-13-11.
Under the hand. None of the shapes for C#maj9 uses an open string, so every one of them is movable; the lowest sits at the 1st fret. 6 of the 11 need a barre. Together they cover the neck from the 1st fret to the 13th.
Shell. Root, third and seventh: the guide tones fix the quality, so the fifth is dropped. That is what leaves a string free, so pick a tension to put on top. C#maj9 has x4354x, 9-8-10-8-x-x, x-x-11-10-13-11.
Drop 2. A close-position chord with the second voice from the top lowered an octave. Sits on four adjacent strings. C#maj9 has x4354x, 9-8-10-8-x-x, 1415xx, xx1521, x-6-10-6-6-x, x-x-10-8-6-9, x-x-11-10-13-11, xx3648.
Drop 3. The third voice from the top lowered an octave. The dropped voice becomes the bass, which is why a string is skipped. C#maj9 has x4x541, 1x152x, x6x668, 9-x-10-8-6-x, 11-x-11-10-13-x, 8x364x, x-8-x-8-13-9, x-15-x-10-14-11.
Rootless. The root left to the bass player. Frees a finger for a tension. C#maj9 has xx1111, xx3544, x6656x, xx6868, x-8-10-8-9-x, x-x-10-10-9-11, x-x-13-13-13-13, x3314x.
Tensions. A shell of C#maj9, root, third and seventh, leaves a string free, and the tones that belong over it are the #11 (F##) and the 13 (A#).
What kind of chord this is. C#maj9 is a major chord — root, major third, fifth. major chords on every root, and what they are for.
The common shapes and the derived voicings come from different places, and both are explained in where the chords and the theory come from.
Shows each chord the way this style would voice it, and marks which of its alternatives belong to the idiom.