D Lydian carries three sharps: F#, G# and C#. That is exactly the key signature of A major, yet the tune never wants to settle on A. It keeps drifting home to D, and the one sharp that a plain D major scale doesn't have, that G#, is responsible for almost everything you'll love about this sound.
The notes of D Lydian are D, E, F#, G#, A, B, C#. Read them out loud and you'll notice every letter from A to G appears exactly once. That isn't an accident. Scales in Western music are written so that each letter name gets one turn. It keeps the notation tidy and makes the scale easy to read on a staff.
That rule explains the spelling of the fourth note. On a piano, the black key between G and A could be called G# or Ab. Here it has to be G#, because we already have an A coming right after it, and we haven't used a G yet. Calling it Ab would leave us with two A's and no G at all, which would be confusing to read and wrong for this mode. So, G#.
The same logic gives us F# (not Gb) and C# (not Db). Once you see that each letter shows up once, the spelling stops feeling like something you memorize and starts feeling obvious.
A whole step is a jump of two piano keys, skipping one in between. A half step is the smallest step in Western music, from one key to the very next, black or white. Lydian follows this pattern:
W - W - W - H - W - W - H
Starting on D:
Look at the start: three whole steps in a row. No other common scale opens like that. Those three steps stretch the distance from D up to G# into an interval called an augmented 4th, one half step wider than the ordinary 4th you'd hear in D major. That stretch is the open, airy quality people describe when they call Lydian dreamy or floating.
Now look at where the half steps land: G# to A, and C# to D. Both lean upward. The C# pulls hard into D, giving you a strong sense of home. The G# leans toward A, which makes the middle of the scale feel bright and lifted rather than settled.
Three sharps normally means A major, and D Lydian really does share every note with A major (A, B, C#, D, E, F#, G#). The difference is which note acts as home. Play the same seven notes but keep landing on D, with a D sounding underneath, and the ear hears D Lydian.
Compare it with D major, which has only two sharps, F# and C#. The single difference is G versus G#. That one raised note, the #4 in the formula 1, 2, 3, #4, 5, 6, 7, is the whole personality of the mode. Sing a D major scale, then sing it again pushing the fourth note up a half step. You'll feel the floor lift slightly.
Stack every other note of the scale and you get these chords:
The chord to notice is E major. In plain D major, the chord on the second note is minor. Here the G# turns it major, and moving between D and E is the classic Lydian sound. Try this on piano or any instrument: D - E - D - E, slowly, then resolve to Dmaj7. Then try Dmaj7 - E - Bm - D. Composers for film lean on exactly this kind of major-chord shift a whole step up to suggest wonder, flight or a wide landscape. The theme to The Simpsons is a well-known example of the Lydian color in popular music.
Be careful with A major. It's a perfectly good chord from the scale, but lean on it too long and your ear may decide A is home, and you're back in A major.
Melodies in D Lydian love to touch the G# and then step up to A, or float on it over a D chord without resolving at all. That unresolved G# hanging over D is the signature moment. Singers and horn players can hold it a beat longer than feels polite; it rarely sounds wrong, just suspended.
Land phrases on D, F# or A to feel grounded. Use E and B as passing color. Save the G# for the moments you want the listener to look up.
Your D roots sit on the low E string at frets 10 and 22, and on the A string at frets 5 and 17. The A-string root at fret 5 is a comfortable home base: play the scale from there and you'll find the G# at the 6th fret of the D string, one fret above where G would be in D major. Strum an open D chord and let that note ring over it.
Hold a low D (a piano key, a drone app, an open string, anything). Over it, sing or play just three notes: F#, G#, A. Go up slowly, stop on G#, and count to four before moving on. Then do the same with G instead of G#. That four-count gap between the two is the entire difference between D major and D Lydian, and once you hear it, you'll recognize it everywhere.
For the bigger picture, read The Lydian Mode.