Interaction term in the field Lagrangian

Can anybody explain me how to imagine interactions between two fields
in the Minkowski space ? For simplicity, if phi is a real scalar field
and the Lagrangian density contains a term of the order phi^4, how can
I understand this term ? If I recall the simple Ising model, the
product of two different spins s_i*s_j gives their interaction strength
which is +1 if the spins are aligned and -1 if anti-aligned; the
interaction occurs at all times and only between different spins. Can
something equally simple be invoked in the field theory ?