Lust haben auf
to feel like
The feeling of wanting to do or have something specific; an inclination or wish.
Ich habe heute Abend Lust auf Pizza.
I feel like pizza tonight.

desire / pleasure
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to feel like
The feeling of wanting to do or have something specific; an inclination or wish.
Ich habe heute Abend Lust auf Pizza.
I feel like pizza tonight.

Careful: 'Die Lust' usually just means 'Lust auf etwas haben' (to feel like something). It is much more general than the English word 'lust', which almost always refers to strong sexual desire.
Hast du Lust, heute Abend ins Kino zu gehen?
Do you feel like going to the movies tonight?
To express what someone feels like, you use the fixed construction 'Lust haben auf' (to feel like) followed by the accusative. This is one of the most common uses of the word.
Die Kinder haben Lust auf ein Eis.
The children feel like having an ice cream.
To express that you do not want to do something, the phrase 'keine Lust haben' (to not feel like it) is very common. Here you use 'keine' and not 'nicht'.
Ich habe heute keine Lust, zu kochen.
I don't feel like cooking today.
The idiom 'nach Lust und Laune' (as one pleases) means that you do something spontaneously and exactly the way you want at the moment, without a fixed plan.
Am Wochenende kann ich endlich ausschlafen und den Tag nach Lust und Laune gestalten.
On the weekend, I can finally sleep in and plan the day as I please.
fun / joke
'der Spaß' and 'die Lust' can both have to do with pleasure. The important difference is: 'Spaß' describes more that something is pleasant, funny, or entertaining. 'Lust' often describes the desire to do or have something, especially in the expression 'Lust auf etwas haben' (to feel like something). So you can have Lust for a movie before you see it. You might then have Spaß during the movie. Learners confuse the words because both sound positive. In practical terms: 'Lust' fits the desire beforehand, while 'Spaß' fits the enjoyment during it.

Nach dem Termin hatte er keine Lust mehr, mit seinem Freund Möbel anzuschauen.
After the appointment, he no longer felt like looking at furniture with his friend.
Die Zugfahrt macht Spaß, weil der Fahrer viele Geschichten erzählt.
The train ride is fun because the driver tells lots of stories.