UPDATE & EDIT: My first answer read as though it was simply trying to establish the use of intellegere when what I meant to make clear is that intellegere is by far and away the most frequently-used verb for this type of exchange (based mainly on Plautus and Terence). Next most used is scire and I have added new material to that section below. I have also expanded on casual ways of asking "do you understand?" and added a new verb, callere.
Looking at casual, conversational exchanges, intellegere is the most frequently used, particularly in the negative. For example:
[quid est?] non intellego
what? I don't understand
Plautus, The Ghost, 475
nam prorsum nil intellego
I don't understand at all
Terence, The Self-Tormentor, 775
non hercle intellego
I don't understand by Hercules
Terence, The Woman of Andros, 190
sed parum intellego quid me velis scribere
but I don't quite understand what you want me to write
Cicero, Letters to Atticus, 380
avunculum meum dixisse: "intellexeras nempe?"
my uncle said, "couldn't you understand him?"
Pliny the Younger, Letters, 5.12
Scire is the next most used verb to say "I don't understand". Here's a great example, with bonus idiom:
quin nec caput nec pes sermoni apparet. nec quid dicatis scire nec me
quor ludatis possum.
Neither head nor foot of your speech is clear. I can't understand what
you're saying or what you're playing at.
Plautus, Comedy of Asses, 729-730
An interesting use of scire is to combine it with noscere to emphasise just how much you don't understand:
nec vos qui homines sitis novi nec scio
I neither know nor understand who you guys are
Plautus, The Braggart Soldier, 450. See also Plautus, Pseudolus, 1210
A similar combination of verbs, this time nescire and capere (the only use of capere I found) occurs in Terence, The Self-Tormentor, 955:
nescio nec rationem capio
I don't know or understand the reason
When asking "do you understand?", again intellegere and scire are both often used (satin intellegis?, iam scis?, for example). But tenere is also frequently seen in both Plautus and Terence: simply tenes? in Phormio 210, or satin haec meministi et tenes? in Plautus, The Persian, 183, are just two examples.
audistin? is attested in Terence, The Brothers, to essentially mean "do you understand?"
Lastly, I came across callere being used to mean "to understand", which was new to me and so I add it here. Examples: callemus probe / we understand well (Plautus, The Little Carthaginian, 575) and docte calleo (Plautus, The Persian, 380).