Tagline
The tagline is the short line under the logo in the header. WordPress calls it the Tagline; it is the same field as Settings → General → Tagline.
Setting it
Appearance → Customize → Site Identity → Tagline.
Keep it to a handful of words. It sits directly under the site name in a header bar that is 60px tall by default, so a sentence will wrap and push the bar out of shape.
A new WordPress install ships with Just another WordPress site. Replace it or turn it off — leaving it is the clearest possible signal that nobody has been through the settings.
Turning it off
Site Identity → Display site tagline — untick it.
The tagline still exists (search engines and feed readers may use it) but it is no longer printed in the header. Emptying the field does the same thing, but keeping the text and switching off the display lets you put it back later without retyping it.
Colour
Site Identity → Tagline color.
Empty — the default — means the tagline follows the header text colour, so a dark header skin gets a light tagline without you picking anything.
It has its own picker rather than sharing the logo’s, because the usual move is to
mute the tagline against a full-strength site name, and one colour cannot do
that. A grey around #777 under a black site name is the classic pairing; on a
dark header, something around rgba-grey — pick a mid grey like #aaaaaa.
The control only appears while Display site tagline is ticked.
Contrast still applies: a tagline muted below roughly 4.5:1 against the header background fails WCAG AA for small text. If you mute it hard, consider raising its size instead — see Text logo size and colours, which governs the site name, and note the tagline reads a notch under it.
With an image logo
The tagline is printed under an image logo exactly as it is under a text one. If your logo already contains a tagline, switch the WordPress one off, or you will have it twice.