Inductance is the property of a closed circuit (circuit meaning a conductor loop) to resist changes in current, specifically due to magnetic flux through the loop. To understand inductance, we first consider when inductance applies. One might begin with Maxwell's equations, and this is indeed the most practical way given that the student is familiar with them. Assuming you are not, for generality in teaching, we will only focus on quantities of interest, and how they relate to each other.
Background
The central quantities in electromagnetism are the electric field (a representation of the electric force), and the magnetic field (a vector which is perpendicular to the 'plane of action' of the magnetic force). When looking at some 2D area in space, where the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ exist and are constant, we can calculate the flux of the fields at (or "through") the surface by taking only the component of $\mathbf{E}$ or $\mathbf{B}$ (or any vector field) which is normal to the surface, at the surface, and multiplying this field value by the area of the surface. For example, the electric flux through a rectangle $2\text{m}\times 2$m square in space, when the electric field has a constant value straight through the surface of $3$ N/C (Newtons per Coulomb), is $3*(2*2) = 12 = \Phi_E$ where $\Phi_E$ is the electric field flux at the surface. To generalize this concept of flux beyond rectangular surfaces and constant fields, we actually take the surface integral of the vector field over the surface.
In practice, the surface is the area in the middle of a circuit loop. We find that, by Faraday's law of electromagnetic induction, if we have a closed loop of wire, and a changing magnetic field $\mathbf{B}$, then there is a current generated in the wire loop which is proportional to the magnetic flux (the product of the magnetic field and the loop area). Note that we are not talking about coils, or resistors, but the entire circuit area.
Inductance
Inductance is easy to define. From Faraday's law, for a circuit loop with a magnetic flux $\Phi_M$, there is an induced voltage $V$,
$$V(t) = -\frac{\partial \Phi_M}{\partial t}$$
Now it's worth noting that a closed loop of wire subjected to changing magnetic flux does not have a definable potential (voltage) in the same way as a circuit with e.g. a battery. The loop is closed, but current flows - the electric field exists, but the voltage is not well-defined. You cannot select two points and calculate or a theoretical voltage drop between them.
Even still, because of the current, a voltage drop is present across e.g. resistors, and this voltage is the induced voltage in the circuit.
Consider, now, a circuit which obeys the relation,
$$V(t) = L \frac{dI}{dt}$$
Where $V(t)$ is the induced voltage, $I$ is the current flowing in the loop, and $L$ is a constant of proportionality. We see that:
$$L\frac{dI}{dt} = \frac{d \Phi_M}{d t}$$
Where the partial derivative has been replaced with a single variable derivative for simplicity, and the right hand side is positive because the EMF under consideration is the back-EMF. Therefore the proportionality constant $L$ is:
$$L = \frac{\frac{d \Phi_M}{d t}} {\frac{dI}{dt}} = \frac{\Phi_M}{I}$$
That is, the ratio of the magnetic flux to the current in the loop. This proportionality constant exists for closed circuits, and is called the inductance of the circuit.
It's important to understand that for a circuit to have inductance, it must be closed (current flows). It's equally important to understand that components in the circuit do not have inductance, but they do contribute to the inductance of the circuit, and this contribution is called the inductance of that component.
The current in a circuit causes a magnetic field around the conductors. The magnetic field, in turn, is responsible for the magnetic flux which determines inductance of the circuit. The other contributing factor is the size of the loop - if the loop is made bigger, the flux increases, and the circuit inductance increases. Similarly, if another circuit (not connected electrically) is nearby, its magnetic field contributes to the magnetic flux in the circuit, and this increases inductance (mutual inductance).
For a circuit with current $I$, if the current remains the same, the magnetic field around the circuit will also remain fairly constant in amplitude. However, moving wires, using shorter or longer wires, coiling wires around each other so their magnetic fields add, all these affect the total magnetic flux through the circuit and thus the inductance. Hence, an inductor is a component which has a geometry that, with a certain amount of current flowing through it, has a known amount of magnetic flux through its area, contributing a known amount of inductance.
Finer Points and Remarks
The property of inductance is the tendency of a closed circuit to affect itself by the current flowing within itself. This follows very simple linear relationships, but involving time derivatives, which give inductance is current-impeding quality. No matter how convoluted the circuit geometry is, all that matters for the inductance is (a) the area enclosed by the circuit, (b) the current flowing through the circuit, and (c) the proximity of current carrying wires to one-another. Calculating inductance is not trivial, but observing what causes inductance can aid the design of simple inductors and the reduction of induction effects.
The reason inductance occurs is the fact that the induced voltage in a circuit is proportional to the magnetic flux changes, and the magnetic flux changes depend on the current flowing through the circuit. The logic is not circular, as it may appear, but rather is a direct result of Faraday's Law. In its common integral form, Faraday's law states that:
$$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot d\mathbf{s}$$
Where $C$ is the path of the circuit, and $S$ is a surface with $C$ as its boundary. Defining the magnetic flux $\Phi_M$ as:
$$\Phi_M = \int_S \mathbf{B} \cdot d\mathbf{s}$$
And the electric potential (or voltage) around a loop $C$ as:
$$V = \int_C \mathbf{E} \cdot d\mathbf{l}$$
Faradays Law reduces to:
$$V(t) = -\frac{\partial \Phi_M}{\partial t}$$
Which is the same equation as we had before, assuming $V(t)$ is the closed circuit potential (zero in the absence of changing magnetic fields).
The only relationship we desire for a circuit, then, is one of the form:
$$V(t) = L \frac{dI}{dt}$$
That is, a linear relationship between change in current and induced voltage. As we have already shown, this is how we come to the quantity of inductance. Inductance is thus a proportionality constant between voltage and changes in current, and describes the tendency of changes in current in a circuit to change the magnetic flux through that circuit. This, through a sort of feedback process, limits the rate at which the current can change.
Edit after rereading this answer to give some further information on practical inductance
Partial, Interior, and Exterior Inductance
The magnetic field around a current-carrying wire is described by the Biot-Savart law, which is an inverse-square law (often somewhat confusingly written as $\mathbf{r}/|\mathbf{r}|^3$). Therefore, the magnetic field contribution of a particular segment of wire is strongest just outside the conductor surface, and falls off very rapidly. This means that most of the contribution to inductance in the overall loop occurs in close proximity to a conductor's surface.
Working along these sorts of lines, we can write the expression for the partial inductance of an open-loop component (like a straight wire, or a coil), even though we cannot strictly define the inductance without a closed loop. Generally, when the magnetic field effects are confined to a small region in a larger circuit, calculating inductance of that region (even if it is not a closed loop) makes for a very good approximation, and gives an indication of the impact that region has on the overall inductance of the loop.
The Biot-Savart law also helps demonstrate why the shape and size of a conductor is a determining factor in inductance calculations. When analyzing a conductor configuration, we often consider a line current (sometimes called a filament), which is centered inside the wire, as the source of the magnetic field. Then, though the Biot-Savart law seems to imply an infinitely strong magnetic field as you approach the center, we only actually integrate the field outside the conductor. The difference between 30 gauge and 22 gauge wire suddenly becomes important, because the smaller the wire, the closer to infinite field strength you can get. In reality of course, currents are not filaments, but rather we have a current density as the source of the magnetic field, with some distribution within the conductor. The fields within a conductor complicate the situation further.
In fact, at low frequencies (specifically, whenever the skin effect can be neglected) we must consider not only the magnetic field around a current-carrying wire to calculate inductance, but also the magnetic field within that wire. Unlike the electric field, which must necessarily be zero in an ideal conductor, the magnetic field can exist within a conductor and (as one might expect from the discussion above) this can have a significant impact. But the interaction of the current density and the magnetic field within the conductor is not so easy to relate to the inductance of the circuit loop. For this reason, the two concepts are often divided into interior and exterior inductance, for inductance within the conductor and inductance outside of the conductor respectively.
Finally, inductance calculations are often complicated when carrying alternating currents because of the skin effect and the proximity effect, both of which can significantly alter the result. These effects are often discussed in RF and microwave design, and in the design of power distribution systems and power engineering. A good reference book (such as Grover's Inductance Calculations) should guide you through various corrections and secondary effects if you want to go further.
