Levene's test is significant. Which test should you use instead?

Unequal variances between groups. Common, not a disaster, and it does not mean your data are unusable.

The short answer

Use Welch's ANOVA instead of the standard one-way F-test, and Games-Howell instead of Tukey for pairwise comparisons. For two groups, use Welch's t-test instead of Student's.

Why

The standard F-test pools variance across groups, which assumes there is one variance to pool. When that is false its error rate is not what it claims, and the problem is worst with unequal group sizes. Welch estimates variance per group and adjusts the degrees of freedom, which is why your df becomes a decimal.

Tukey's HSD carries the same pooled-variance assumption. Games-Howell is its unequal-variance equivalent and already corrects for multiple comparisons, so do not apply a further correction on top.

The honest caveat

Testing for equal variances and then choosing your test based on the result is itself a two-step procedure that distorts the overall error rate. Many statisticians now recommend defaulting to Welch regardless, since it performs almost as well when variances are equal and much better when they are not. Worth knowing before a reviewer raises it.

In Plotara

Levene's test runs as part of the ANOVA. When variances differ the app reports Welch's ANOVA with Games-Howell automatically, and the Methods text names which version was used.