Judaism accepts converts with open arms: So yes, if this is what you would like to do, you are welcome to convert to Judaism. Once someone converts, they are considered to have been Jewish from birth. You would be considered as Jewish as everyone else.
It is important to stress that since conversion requires a complete change on beliefs, it is highly respected in the Jewish community (it is much harder to fully believe in something that is different from how you were raised).
Also, take into account a few things: First and foremost, is that conversion is not an easy process. You would be taking on many rituals, giving up others and essentially changing your lifestyle forever. Secondly, it can take up to seven years in some cases before you actually get to convert.
The reason for this is because Judaism wants to discourage spontaneous choices in this matter. One should know and understand every aspect of the laws they are agreeing to, and should have all their questions answered as they learn.
You also need to ask yourself:
Why do you want to be Jewish?
Why does it appeal to you?
Can you handle observing the commandments?
What does it mean to be Jewish?
If you are convinced about knowing more, speak with your a local Rabbi at a synagogue. I would also suggest that you speak with a Rabbi from all three movements (Orthodox, Conservative, and Reform), in order to see if anything fits well with you.
So, research a bit first, and then speak with your local Rabbi.
Here are some Jewish informational sites, to help you learn a little bit more: