What errors can occur during DNA replication?
There are many way in which DNA can be damaged. As already pointed out in the comment by @skymninge, the Wikipedia page on DNA repair, as well as the mutation page detail some of the things that can go wrong. 

You say:


  If G goes with C and A goes with T, I don't see how that part can mess up.


This, however, would imply that the four bases are completely different so that mispairing cannot exist. This is (may I add luckily?) not true.
Indeed the chemical structure of the bases is very similar, and changes between one base and another (transversions and transitions) are common.



These can result for instance from exposure to external agents such as ionizing radiation and alkylating agents, or from exposure to endogenous products such as reactive oxygen species.

Furthermore, the enzyme synthetizing DNA, called DNA polymerase, can insert the wrong base, although most of these errors can be corrected by its "proof-reading mechanisms". However, it is estimated that


  the frequency at which human DNA undergoes lasting, uncorrected errors range from $1 * 10^{-4}$ to $1 * 10^{-6}$ mutations per gamete for a given gene.  


Source: Nature Scitable

So, in summary, the system is not perfect, and mutation can be introduced even endogenously. This may seem bad, but, at least for relatively small rates of mutation, is not, as it allows change in the population, which is the basis for evolution.