Take the major scale and raise its only troublemaker: the 4 goes up a half step, and the textbook avoid tone becomes the prettiest note in the room. Nothing pulls anywhere — the scale's tritone now sits between 1 and #4 instead of aiming at home — so Lydian doesn't resolve, it hovers. Summon it by parking on 1Δ⁷ and visiting the major chord on 2: that chord carries the #4, and nothing in plain major can. Film scores and dream pop live on this two-chord vamp because it sounds like the moment before something wonderful.
The #4 — it floats a whole step above the 3rd where plain major's 4 grinds a half step, and the tonic sounds airborne.
Your tonic is the 4 of somebody's major scale — the parent key sits a perfect 5th up. F Lydian is C major's notes with F as home.
The #4 is tonic-flavor only — the moment a real 4 chord or a 2- arrives, the harmony has voted for plain major and holding the #4 is just a wrong note. And don't cadence: one honest 5 → 1 snaps the spell back to Ionian.
The Simpsons main theme opens on it; Joe Satriani's 'Flying in a Blue Dream' solos in it; film composers reach for the 1–2 shimmer whenever wonder is on the call sheet.