In layman language, differential equation is an equation that relates functions and quantity with rates at which unknown functions are changing.
Any equation with derivatives, differentials, or partial derivatives is a differential equation.
An example of a differential equation[Simple Harmonic Motion]:
F=d/dt[p] where p is the momentum and F is the net force.
Hooke's law says: F=kx
Setting these two equal, we get, and assuming mass to be constant we get:
kx = mx'' which is a 2nd order linear homogenous ODE with constant coefficients.
Now, notice that with the same spring constant k, and mass m, the motion of an object on the spring also depends on the initial position and initial velocity. The diff eq just models the general case and gives us a family of solutions. If we want to solve for a particular case, we just need those values. That is what initial conditions are. The initial conditions here can be x'(t0)=v0 x(t0)=x0 for some time t0 <Note by t0, I mean t subscript 0 and so on>.
Hopefully, I give you a better insight as to what a differential equation is physically. I by no means intend to be rigorous by my definitions.