I performed a one way between groups anova and got a significant result for one of the variables. However, the tukeys test was not significant. Why is this not significant and what post hoc tests should I be running to look at the significance of this variable?
ANOVA and posthoc tests have nothing to do with each other!
ANOVA tests the "exploratory value" of a predictor in a model (typically a factor with more than two levels). It answers the questions how likely we can expect a reduction in the residual variance when there was no association of the presumed predictor with the response.
Posthoc tests are apriori unspecified tests, so to say tests *after seeing* the data and deciding then which comparisons might be interesting. They use pooled variance estimates and control the FWER or the FDR for the family of tests. There are few special cases of posthoc tests that do not efficiently control the FWER. They need a kind of "protection" by a "significant" ANOVA. Most famous example is Fisher's LSD, but this is restricted to 3 groups. Tukey does not require any protection and it keeps the FWER.
In general, ANOVA and posthoc tests answer considerably different questions. Your observation is thus not puzzeling at all.
ANOVA tests for differences in group means when you have 3 or more groups. Statistically significant result implies some of the groups have different means or in other words some of the groups dont have equal means. Tukeys test or any post-hoc tests shows result of which of the groups is significant. Further, such post-hoc tests control the experiment-wise error rate by reducing the statistical power of the comparisons. When you look at the graph of Tukey's simultaneous 95% Confidence intervals for the groups, the group where the CI does not include a 0 is the one which has statistically different means.
Reasons, why that may happen, could be a small group size or a high number of factor levels. A p-value equal or near the significant level in the ANOVA could also explain why the Tukey HSD showed no significant difference. Of course there are other explanations as well.