The FEM method for transient problems typically uses the method of lines, i.e. the spatial discretization is decoupled from the time discretization:
\begin{equation}
u^h(x,t) = \mathbf{\Phi}(x)^T \, \mathbf{U}(t)
\end{equation}
where $\mathbf{U}(t)$ is the vector of nodal quantities, assumed as unknown functions of time. Under this assumption the space-time PDE's in $(x,t)$ are reduced (discretized) to ODE's in $t$ alone using the usual FEM machinery for static problems.
As already pointed out by other answers, we speak of explicit or implicit FEM with reference to the time integration scheme of these ODE's.
With reference to continuum mechanics problems (without damping), we end up with a system of ODE's like
\begin{equation}
\mathbf{M} \ddot{\mathbf{U}}(t) + \mathbf{F}_\text{i}(\mathbf{U}(t)) = \mathbf{F}_\text{e}(t)
\end{equation}
where $\mathbf{F}_\text{i}$ and $\mathbf{F}_\text{e}$ are the internal and external nodal equivalent forces. For linear problems $\mathbf{F}_\text{i}(t) = \mathbf{K}\, \mathbf{U}(t)$.
At the risk of over-simplifing let us assume that in an explicit scheme you just need to solve for $\ddot{\mathbf{U}}(t)$
\begin{equation}
\mathbf{M} \ddot{\mathbf{U}}(t) = -\mathbf{F}_\text{i}(\mathbf{U}(t)) + \mathbf{F}_\text{e}(t)
\end{equation}
which is trivial if the mass matrix is lumped. On the contrary in implicit methods you need to solve the (non)-linear equations $\mathbf{F}_\text{i}(\mathbf{U}(t)) = \mathbf{b}$.
To fully answer your question: explicit/implicit refers to the solution of the ODE system and not the nature of the mass matrix. Of course every reasonable implementation of an explicit scheme requires the mass matrix to be lumped, otherwise the advantages of the method are lost in the solution for $\ddot{\mathbf{U}}(t)$. On the contrary for implicit schemes you can have both lumped and consistent mass matrices.