Rats symbolizes good luck and "plenty", because if one has rats, that means there is enough stored grains around in the house to have them.
In the Zodiac Compass the rat represents North, which is also the first year in the cycle. The rotating cycle of twelve animal signs was a folk method for naming the years in traditional China. The animal signs rotates in an established order every twelwe years. For example 1976 was the Year of the Dragon, 1977 was the year of the Snake. After twelve years it will be the year of the Dragon and the year of the Snake, again and so on.
The reason why the Rat is the first animal in the cycle is explained in Chinese legend. The twelve animals were one day quarreling on who was to head the cycle of years, so they decided to settle the matter by swimming across a river; whoever was to reach the opposite bank of the river first would be the first in the cycle, and the rest of the animals would receive their years according to their finish.
All the twelve animals gathered at the river bank and jumped in. The ox took the lead but unknown to the ox, the rat had jumped upon his back. So as the ox was about to jump ashore the rat jumped off the ox's back and won the race. The pig, who was very lazy, ended up last. That is why the rat is the first year of the animal cycle, the ox second, and the pig last.