When you or somebody else casts a prolonged effect spell on you, there's an icon in the upper or lower part of the screen respectively, but no explanation what this or that icon means. Well, you can set specific icons for your own spells (though not for potion or item effects), which partially solves the problem, but spells suffered from monsters are rather vague. Is there a way to determine which exactly spell effects are applied to the character?

No real way to tell, except for what seems to be happening to your character.

Usually, it's one of three things--
1. You're silenced and you can't cast spells until it wears off. However, I think the game says "You are silenced." if that happens, so you'll know.
2. It was a paralysis attempt (if you can move then it failed)
3. Someone cast a spell on you that's slowly draining your magic points away.

I'm not sure if there are any other "enduring" effects but I'm fairly certain that's it. Eventually, you'll remember what creature does what.

Yea, if it's not one of the obvious effects like the mentioned paralysis or silence, then your best bet is to open your character sheet and look for red numbers.