EDIT: I had not realized this was about the data format. You could use
import pandas as pd
import scipy
two_data = pd.DataFrame(data, index=data['Category'])
Then accessing the categories is as simple as
scipy.stats.ttest_ind(two_data.loc['cat'], two_data.loc['cat2'], equal_var=False)
The loc operator accesses rows by label.
one sided or two sided dependent or independent
If you have two independent samples but you do not know that they have equal variance, you can use Welch's t-test. It is as simple as
scipy.stats.ttest_ind(cat1['values'], cat2['values'], equal_var=False)
For reasons to prefer Welch's test, see https://stats.stackexchange.com/questions/305/when-conducting-a-t-test-why-would-one-prefer-to-assume-or-test-for-equal-vari.
For two dependent samples, you can use
scipy.stats.ttest_rel(cat1['values'], cat2['values'])