What is resolution?
Resolution in this context refers to the conversion of an analog voltage to a digital value in a computer (and vice versa). A computer is a digital machine and thus stores a number as a series of ones and zeroes. If you are storing a digital 2bit number you can store 4 different values: 00, 01, 10, or 11. Now, say you have a device which converts an analog voltage between 0 and 10 volts into a 2bit digital value for storage in a computer. This device will give digital values as follows:
Voltage 
2Bit Digital Representation 
0 to 2.5
2.5 to 5
5 to 7.5
7.5 to 10

00
01
10
11

So in this example, the 2bit digital value can represent 4 different numbers, and the voltage input range of 0 to 10 volts is divided into 4 pieces giving a voltage resolution of 2.5 volts per bit. A 3bit digital value can represent 8 (2^{3}) different numbers. A 12bit digital value can represent 4096 (2^{12}) different numbers. A 16bit digital value can represent 65536 (2^{16}) different numbers.
It might occur to you at this point that a digital input could be thought of as a 1bit analog to digital converter. Low voltages give a 0 and high voltages give a 1.
In the case of the LabJack U12, a singleended analog input has a voltage range of 10 volts to +10 volts (20 volt total span) and returns a 12bit value. This gives a voltage resolution of 20/4096 or 0.00488 volts per bit (4.88 mV/bit).
What does it mean to say a device is 12bit, 16bit, or 24bit?
When you see analog input DAQ devices from various manufacturers called 12bit, 16bit, or 24bit, it generally just means they have an ADC (analog to digital converter) that returns that many bits. When an ADC chip returns 16 bits, it is probably better than a 12bit converter, but not always. The simple fact that a converter returns 16bits says little about the quality of those bits.
It is hard to simply state "the resolution" of a given device. What we like to do, is provide actual measured data that tells you the resolution of a device including typical inherent noise.
If you look at a device called "24bit" just because it has a converter that returns 24bits of data per sample, you will find that it typically provides 20 bits effective or 18 bits noisefree (like the UE9Pro). The U6Pro and T7Pro provide some of the best performance around from a 24bit ADC, and they do about 22 bits effective or 20 bits noisefree. You will see with these devices we might mention they have a 24bit ADC (as that is what people look and search for), but we try not to call them "24bit" and try to stick with the effective resolution.
Another interesting thing about your typical 24bit sigmadelta converter, is that you can look at them as only having a 1bit ADC inside, but with timing and math they can produce 24bit readings:
http://www.maximic.com/appnotes.cfm/appnote_number/1870/
Hardware with a 24bit ADC
Hardware with a 16bit ADC or less