Negative T Stat?

I'm used to getting very large t stat values and know that the larger they get, the more significant my results are.

However, what does a negative t stat really mean? Something like t = -2?

Using a two sample t-test assuming unequal variances, I compared two groups (sample size 100 each) with means that are about the same (only 0.04 difference) and hypothesized a mean difference of 0. My p-value is about 0.02 (one tail) and 0.04 (two tail).

Does this mean that the null hypothesis (i.e. mean difference = 0) can be rejected at the 0.02 level?

Also, which result should I take? The one tail or two tail test and why?
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