If you look carefully at the words gespannt and entspannt in the sentence below, you'll notice that gespannt is a participle (of spannen), whereas entspannt is an adverb.
Dann werden die Pferde vor die Kutsche gespannt und man kann ganz entspannt durch Dahlem fahren.
Then the horses are hitched up in front of the coach and one can ride very leisurely through Dahlem.
Captions 39-40, Berlin - Domäne Dahlem - Part 1
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However, you have probably also seen gespannt used as an adjective, and also perhaps come across spannend or verspannt. Let's take a look at some examples and clarify the meanings of all of these words.
Above, entspannt is used as an adverb to mean "leisurely" or "in a relaxed way." As an adjective it means "relaxed," or you might see it as the verb entspannen, which means — you guessed it — "to relax."
Aber es entspannt mich. -O ja.
But it relaxes me. -Oh yes.
Caption 15, Kolkhorst - Konzert
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Und das ist sehr, sehr, sehr entspannt im Gegensatz zu München, Hamburg, Berlin.
And that's very, very, very relaxed in contrast to Munich, Hamburg, Berlin.
Caption 19, Fernsehmoderatorin - Sonya Kraus
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Spannend is an adjective used to describe something that is "exciting," "interesting," or "intriguing."
Auf der Berlinale zu laufen ist schon wahnsinnig spannend.
Attending the Berlinale, indeed, is insanely exciting.
Caption 7, Berlinale - Schauspieler Jürgen Vogel
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Ich finde es sehr spannend, wie viele Facetten so ein Film hat
I find it very exciting how many facets a film like this has.
Caption 48, Filmtrailer - Der kleine Rabe Socke
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Be aware: spannend almost never describes people. It means "exciting" and NOT "excited," which is the meaning of the adjective gespannt.
Ich bin mal gespannt, wie es klappt. Ich hoffe, es gefällt euch.
I am excited to see how it will go. I hope you like it.
Caption 41, Eva Croissant - Interview - Part 2
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Two other words you should know are angespannt and verspannt. They both translate as "tense" in a more negative sense, and often refer to the body rather than the mind.
Bauch ist leicht angespannt. Wenn wir hochkommen: ausatmen...
The belly is slightly tensed. When we come up, breathe out...
Caption 34, Workout mit Erik - Übungen für Arme, Beine, Po, Rücken
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Genau da, wo er so verspannt ist, da muss er ziehen.
exactly where it is so tense, it has to stretch there.
Caption 32, Nackenschmerzen, steifer Nacken, steifer Hals - Übungen gegen Nackenverspannungen
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Further Learning
Make sure that you understand all related words and can identify what part of speech they occupy in sentences using context clues. This includes the verbs spannen, sich entspannen, (sich) verspannen, and anspannen. There are many examples available on Yabla German.