What is a Lagrangian and what is the Hamiltonian? How are they related?

The energy-momentum tensor $T^\mu_\nu:=j^\mu_\nu=\partial_\nu\phi\frac{\partial \mathcal{L}}{\partial(\partial_mu \phi)}-\eta^\mu_\nu \mathcal{L} ~\text{and}~ \partial_\mu T^{\mu\nu}$(Not symmetric)

Relation: $P^0=\int d^3x T^{00}=\int d^3x \mathcal{H}=H$

with canonical momenta: $\mathcal{H}=\pi\partial_0\phi -\mathcal{L}$