The modern Chinese character for sheep or goat is 羊. It does not look like a sheep / goat any longer. Its ancient scripts might help us to reason its meaning, which is shown below (image taken from http://vividict.com/):
|Evolution history of 羊|
Looking at its Oracle script and , or its Bronze script , we can see 羊 is the drawing of a head of sheep or goat. It is because of the same reason that sheep or goat is also a mainly used sacrifice offered to gods.
|Sacrifice offered with a head of a goat|
We can also see that at certain age of Bronze script whole body of 羊 was drawn to represent sheep or goat, which was eventually abandoned.
So 羊 means sheep or goat. As Chinese character is a drawing, it can be used to represent any animals looks like a sheep or goat. With this understanding, we will no longer be surprised to see Chinese can use either sheep or goat for their year of Chinese zodiac 羊.
As 羊 is always considered as a quiet, gentle and peaceful species, especially it is always used as sacrifice to gods for blessing, 羊 itself has the meaning of peace, gentle, auspicious and being blessed, especially when 羊 was used as a component to create other Chinese characters.
Now it is time to enjoy a Chinese calligraphy with 羊 inside: