A Hermitian (or symmetric) matrix $A$ is positive definite iff all its eigenvalues $\lambda_i$ are positive; let's restrict, for simplicity to the case of matrices with real entries.
A way to ensure that a matrix of interest is positive definite is therefore to check that its minimum eigenvalue is bounded away from zero and positive. In this regard, lower bounds on eigenvalues could be of some help: one does not need to compute eigenvalues explicitly; if he/she can show that the lower bounds are strictly positive, then positive semidefiniteness is ensured.
Lower bounds can be in terms of objects which are more tractable, such as norms of the matrix elements, determinants, etc. so that showing that a lower bound is positive may be an easier task than proving such property directly for eigenvalues.
Are there lower bounds for the eigenvalues of Hermitian matrices typically used for this purpose?