When it comes to processing speed, there are mainly two things which matter, the processing speed and the number of cores. Speed is measured in GHz. These days most Androids have a 1.2GHz-1.7GHz as their speed, which is pretty good for a phone. However the number or cores is extremely important. The number or cores is literally the number of processors, so if your Android has a dual core processor than it means there are two cores; and that means your phone's processor can do two things at the same time, at the given speed in GHz.
Also note that a dual core 1GHz processor does not mean that your processor is 1 multiplied by 2 = 2GHz. It just means that your 1GHz processor can process two things at the same time.
General meaning apart, the higher the better; though a faster processor does consume more battery.
A high end Android would have a 1.4 - 1.7 GHz quad core processor. Though dual core would also be just as acceptable. These phones can run almost ANYTHING that an Android is capable of without lag. And can also handle a considerable amount of multi tasking. But even these processors have their limits, if you have too much stuff running at the same time even that will lag.
A mid range smartphone would have a 1.2 GHz dual core processor, which is still fast. It can run pretty much everything a high end Android can, only with less multi tasking capabilities.
The brand doesn't really matter that much, qualcomm, samsung, whatever. When looking at the processor, just look at the speed (the GHz) and the number of cores, don't worry about the brand that much.