Quantization simply means that the values are not continuous but are rather discrete. For example, weight is continuous because you can weigh anything up to any desired decimal point (the only limit is your instrument). On the other hand, the number of people is discrete; I can't have 1.2 persons - it's either 1 or 2.
Specifically, by saying that charges are quantized, we're saying that charges cannot attain just any value (unlike weight -- if you think of any number, that could be the weight of some arbitrary object). Charges are quantized because the charge of any object (ion, molecule, etc.) are multiples of a fundamental quantity -- we can say that any charge can be expressed as `ke` , where `k` is an integer, while `e` is the fundamental unit of charge - or the elementary charge.
The value of `e` is the charge of the electron/proton (they only differ in sign) which is approximately `1.602 times 10^(-19)` coloumbs. Charge quantization, then, means that charge cannot take any arbitrary values, but only values that are integral multiples of the fundamental charge (charge of proton/electron). For example, in a hydrogen ion, we usually denote it with a positive sign to indicate that there's one proton more than there are electrons. The positive sign actually corresponds to one fundamental charge. Hence, we can have charges that are 2e, -5e, 10e, and 6e but not 1/2 e, 1/5e, and other non-integer values. This is the quantization of charge.
Note, however, that in the case of quarks, the quantization is different - they are in multiples of 1/3 the fundamental charge (`e/3` ). However, they cannot be isolate. Particles that can be isolate have charges that are integral multiples of the fundamental/elementary charge.