This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for the first eigenvalue are given. These bounds are guaranteed, fully computable, and converge with the optimal speed to the exact eigenvalue. They are valid under explicit, a posteriori conditions on the computational mesh and on the approximate solution; we also need to assume that the approximate eigenvalue is smaller than a computable lower bound on the second smallest eigenvalue, which can be verified in many cases of practical interest by including the computational domain into a rectangular parallelepiped or a d-sphere. Guaranteed, fully computable, and polynomial-degree robust bounds for the energy error in the approximation of the first eigenvector are derived as well, under the same conditions. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory, and no convexity/regularity assumption on the computational domain/exact eigenvector(s) is needed. An improvement of the multiplicative constant appearing in our estimates is possible when an elliptic regularity assumption is satisfied with known constants, typically for convex two-dimensional domains. The application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided, along with a numerical illustration on a set of test problems. Key words: Laplace eigenvalue problem, a posteriori estimate, eigenvalue error, eigenvector error, guaranteed bound, conforming finite element method