For problems you have in the future like this that don't involve coding you should post at https://datascience.stackexchange.com/.
This depends on a few things. First of all, how important is this variable in your exercise? Assuming that you are doing classification, you could try removing all rows without with NaN values, running a few models, then removing the variable and running the same models again. If you haven't seen a dip in accuracy, then you might consider removing the variable completely.
If you do see a dip in accuracy or can't judge impact due to the problem being unsupervised, then there are several other methods you can try. If you just want a quick fix, and if there aren't too many NaNs or categories, then you can just impute with the most frequent value. This shouldn't cause too many problems if the previous conditions are satisfied.
If you want to be more exact, then you could consider using the other variables you have to predict the class of the categorical variable (obviously this will only work if the categorical variable is correlated to some of your other variables). You could use a variety of algorithms for this, including classifiers or clustering. It all depends on the distribution of your categorical variable and how much effort you want to put it in to solve your issue.
(I'm only learning as well, however I think thats most of your options)