If you have a weak instrument then the bias of the IV estimator can be large and in some cases it can even be bigger than the bias of the OLS estimator. With their tabulated values Stock and Yogo first fix the largest relative bias of the two stage least squares estimator (2SLS) relative to OLS that is acceptable. In this sense the test answers the question: can we reject the null hypothesis that the maximum relative bias due to weak instruments is 10% (or 5%, etc.).
The critical values then depend on this acceptable bias (a lower acceptable bias means that your instrument has to achieve a higher first stage F-statistic), the number of endogenous regressors and the number of exclusion restrictions. As an example, if you set the maximum acceptable bias to 0.05 (i.e. we tolerate a bias of 5% relative to OLS), and you have one endogenous variable and three instruments, the critical value is 13.91, so your instrument is not considered weak if its first stage F-statistic is larger than that. The problem is that these critical values only work if you have at least two overidentifying restrictions. In your case with one endogenous variable you need at least three instruments.
With one endogenous variable the Cragg-Donald test should give you a similar result as Stock and Yogo. This test differs from the previous one if there are several endogenous variables for which you will have multiple first stages. Anderson's canonical correlation test works similar to Cragg-Donald with the difference that Anderson's CC is a likelihood ratio test whilst Cragg-Donald is a Wald statistic but both tests are applicable with one endogenous variable and one instrument. However, in the end Stock Yogo, Cragg-Donald and Anderson all rely on an iid assumption on the errors. If you used heteroscedasticity robust standard errors for instance these tests will not work but the Kleinbergen-Paap test is robust against violations of the iid assumption. It also works with one endogenous variable and one instrument as long as the model is identified. There is a nice discussion on these tests in these notes by Baum (2007). Other robust tests for weak instruments are also offered in Stata's
If you end up with a weak instrument you can use the conditional likelihood ratio test by Moreira (2003) in order to perform weak instrument robust inference. A paper by Andrews et al. (2008) shows that the CLR test is approximately optimal. Weak robust instruments regressions are available for instance in Stata's