Wow, there's lots to deal with here. I'm assuming this is homework, so my first advice is that you catch up on the expected reading!
Second, the "significance level" has a different meaning depending on whether you are working within the dichotomous hypothesis testing framework of Neyman or the significance testing framework of Fisher. In the former case the significance level is an unfortunate phrase indicating the alpha level or 'size' of the test, and it might be 0.05 as you indicate, but only if that level is explicitly decided prior to the data being analysed. In the latter case the significance level refers to the observed significance level, the p-value. (That distinction is not always observed by introductory stats texts and may be unknown to some instructors.)
Next, if you are doing a Student's t-test then the test statistic is t. I expect that the question would be answered by an explanation of the meaning of the t statistic and how it is calculated for the particular experiment.
Next, the t-test that you have calculated using R is a Welch's variant. It relaxes the assumption of the original Student's t-test that the variances of the populations are equal. The way it is calculated leads to fractional degrees of freedom. For a conventional Student's t-test your degrees of freedom would be n1-1 + n2-1.
Finally, the distribution that you have been asked for is the distribution of t under the assumption that the parameter of interest (mean difference) is equal to the null hypothesised value (usually zero, but it can be any value desired). The tabulated critical values of t that you consulted will not suffice.
