jemandem ist übel
to feel nauseous
To have the feeling of nausea, as if you might have to vomit.
Nach der langen Autofahrt war mir furchtbar übel.
After the long car ride, I felt terribly nauseous.

nauseous / bad
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to feel nauseous
To have the feeling of nausea, as if you might have to vomit.
Nach der langen Autofahrt war mir furchtbar übel.
After the long car ride, I felt terribly nauseous.

To express the feeling of nausea, German uses a fixed construction with the dative: 'jemandem ist übel' (someone feels nauseous). The person who feels sick is always in the dative.
Dem Kind wurde auf der langen Busfahrt übel.
The child felt nauseous on the long bus ride.
A typical mistake is using the nominative (*Ich bin übel*). That is wrong and would mean that you are a bad person. The correct form is always: 'Mir ist übel' (I feel nauseous).
Wenn ich zu viel Süßes esse, wird mir schnell übel.
If I eat too many sweets, I quickly start feeling nauseous.
The idiom 'jemandem etwas übel nehmen' (to hold something against someone) is a fixed structure. The person you resent something from is in the dative, and the action or statement that bothers you is in the accusative.
Er hat mir meine ehrliche Kritik sehr übel genommen.
He really held my honest criticism against me.
When 'übel' (bad/nasty) comes before a noun that describes an action or quality, it often has a moral meaning. Typical combinations are 'eine üble Tat' (a bad deed), 'üble Absichten' (evil intentions), or 'üble Gerüchte' (nasty rumors).
Über ihn werden in der Firma üble Gerüchte verbreitet.
Nasty rumors are being spread about him at the company.
bad
'schlecht' and 'übel' can be similar in some situations. Especially with nausea, you can say 'mir ist schlecht' or 'mir ist übel' (I feel sick / nauseous). 'übel' puts stronger emphasis on the feeling that you might have to vomit. 'schlecht' is more general and can also describe quality, morality, or spoiled food. 'übel' can also mean morally bad or very unpleasant in terms of smell and taste, but it often sounds somewhat stronger or more specific. Learners may confuse the words because both fit when someone feels unwell. In everyday language, 'mir ist schlecht' is very common and general, while 'mir ist übel' is more precise for nausea.

Meinem Bruder ist nach der Busfahrt übel.
My brother feels nauseous after the bus ride.
Nach dem alten Joghurt ist mir schlecht geworden.
After eating the old yogurt, I felt sick.