In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!
Technically a synonym for function, "operator" in physics almost always means either
A linear mapping between two function spaces (defined on, say, $\mathbb R^n$), or
A square matrix (representing a linear mapping from a vector space to another according to a certain basis).
Some examples:
The so-called momentum operator, $D \colon C^1\to C^0$ such that $f(x)\mapsto -i\hbar \nabla_xf(x)$ defines an operator $D$ as typically thought of in physics, and
The Fourier transform $\mathcal{F}\colon{L}^1(\mathbb R^n)\to{L}^\infty(\mathbb R^n)$.
In quantum-mechanics, observables are Hermitian operators acting on the hilbert-space of quantum states, together with a "recipe" for physically interpreting them, namely: straight after an observation, the quantum state $\psi$ is projected onto a random eigenfunction $\psi_j$ of the operator $F$ with probability (or probability density) $|\langle\psi|\psi_j\rangle|^2$, and the $M^\text{th}$ moment of this probability distribution given quantum state $\psi$ is $|\langle\psi|F^M\psi\rangle|^2$.
Another common physics concept of an operator, more in line with its mathematical meaning, is a state transition or time-evolution operator in quantum mechanics $U(t) = \exp(i\hat{H}t)$ that maps a quantum state $\psi$ into its value $U(t)\psi$ a time $t$ later in the Schrödinger picture, where $\hat{H}$ is the hamiltonian (energy) observable.
