I can see what the other answers are trying to say, but they're not quite right. There IS something very much like a wave function in QFT, and it does describe those things. There also is a Schrodinger equation.
Since you are a layman, I have to ask, to what extent do you know the basics of QM? Are you aware of the Schrodinger equation and the bra-ket formalism? If yes, then I can explain to you very simply what QFT is: exactly the same thing as QM, except that where in QM you typically have observable operators of, say, position $\hat{x}$, and a wave function that depends on time, in QFT position is no longer an operator but a parameter just like time, and the role played by $\hat{x}$ in QM is played by fields $\hat{\phi}(x)$.
Notice how the field depends on the parameter $x$ (which is NOT an operator, like in QM), while in QM the position operator $\hat{x}$ doesn't depend on anything. This is what people mean when they QFT has infinite degrees of freedom, to specify a field completely you need to specify it in every point of space. Conversely, notice that if you got rid of all spacial parameters in QFT and retained only time, you would get something that looks very much like ordinary QM, with $\hat{x}$ being a "field". That's why people say QM is QFT in 1 dimension (that dimension being time).
What is the wavefunction then? Well, it's kinda different from ordinary QM because it's an abstract vector in Hilbert space, you can't just write its value as a number in some place, like you can with QM. It's a bit more complicated than that because it carries a lot more data (remember the infinite degrees of freedom). But what is relatively simple is to produce the wavefunction that corresponds to, say, an electron and a positron as excitations of a quantum EM field at some distant time in the past, produce the wavefunction corresponding to a photon in the distant future, and then find the probability that one state evolves into the other, ie that the electron-positron pair annihilates into a photon (this way of treating problems is called scattering theory). Performing that calculation is how you get the Feynman diagrams.
Bare in mind in QFT people typically use the Heisenberg picture, in which the states ("wavefunctions") remain the same and the fields (or position/momentum operators in QM) evolve with time. I've explained it with the Schrodinger picture because it is probably what you are familiar with. But really both are exactly the same, one transforms into the other by a simple mathematical transformation.
