an initial condition
Suppose you want to predict how a hot cup of coffee cools, or how a plucked guitar string moves. The equation alone — say the heat equation u_t = k u_xx, or the wave equation u_tt = c^2 u_xx — only tells you the rule the system obeys as time runs forward. It cannot tell you where to start. An initial condition is the photograph of the system at the very first instant, time t = 0: it pins down the value of the unknown function (and, for some equations, its rate of change) all across space at that one moment, so that the equation has a definite future to run.
Concretely, for an equation that is first order in time, like the heat equation, an initial condition prescribes u(x, 0) = f(x): the whole temperature profile along the bar at t = 0. For an equation that is second order in time, like the wave equation, you need two pieces of starting data, because a second-order ODE in time needs both a position and a velocity to be determined: u(x, 0) = f(x), the initial displacement, and u_t(x, 0) = g(x), the initial velocity. The number of time-derivatives you must prescribe matches the order of the equation in time — exactly as an ODE of order n needs n initial values. Once the photograph (and possibly the initial velocity) is fixed, the PDE propagates that data forward.
The reason initial conditions are unavoidable is that a PDE, unlike an algebraic equation, does not have a single answer: its general solution involves arbitrary functions, not just arbitrary constants. The heat equation is satisfied by infinitely many temperature histories; the initial condition is what selects the one that matches your actual experiment. A common confusion is to think the initial condition must itself solve the equation — it need not. It is free data, an input you supply; the solution is the output the equation builds from it as time advances.
For the heat equation u_t = k u_xx on the bar 0 <= x <= L, prescribe the initial temperature u(x, 0) = f(x) = sin(pi x / L). The equation then evolves this single bump so it decays smoothly toward zero: u(x, t) = e^(-k (pi/L)^2 t) sin(pi x / L). The initial condition chose the shape; the equation chose the decay.
The initial condition supplies the t = 0 profile; the PDE supplies everything afterward.
How many initial conditions you need is set by the order in time, not the order in space: the wave equation is second order in time, so it needs both an initial displacement and an initial velocity, while the heat equation, first order in time, needs only the initial profile.