I'm sorry is this is too obvious, but I'm having a hard time trying to find a distribution for my data. It is clearly not a normal distribution. It does not seem to be skewed, but seems to have fat tails. Is that right? I thought a Student distribution would be the closest but I'm not sure. I need the distribution to be able to fit a GARCH model. Thanks in advance!
You can't readily use the raw (marginal) response to choose a conditional distribution for a GARCH model - if the model is approximately correct, then the marginal distribution of the data will be a scale mixture that will look heavier tailed than the conditional distribution you need to choose. [This is a similar problem to trying to choose an error distribution with a regression model from the raw response, except in that case you have a location-mixture.]
Since I don't have your data, I don't know what model you should choose. Generally speaking, normals are not heavy tailed enough (and log returns tend to be slightly left skew).
Sometimes people try t-errors, but you can't estimate the d.f. from the above marginal information (indeed d.f. in the t are hard to estimate in any case; it may be better to choose a d.f. somewhat arbitrarily). You could for example assume something (low d.f like 5 or 7 seem to be fairly common choices, but it's not really my area) and then look at whether it reasonably approximates the estimated errors in the model (eg. via a QQ plot).