Definition: Electromotive Force
The electromotive force, denoted by $\mathcal E$ is a generalization of the potential difference. Instead of just looking at the the electric field, it looks at the force per charge from the lorentz force: $$\mathcal E = \int ((\mathbf v \times \mathbf B) + \mathbf E )\cdot d\mathbf r.$$
For a fixed loop moving in a temporally constant magnetic field (not necessarily spatially), integrating the expression above (check this as an exercise for yourself!) gives, $$\mathcal E = \int (\mathbf v \times \mathbf B) \cdot d\mathbf r = -\frac{d\Phi_B}{dt}.$$ More generally, for a non-temporally constant magnetic field, we have Faraday's law.

Theorem: Faraday's Law
In the case of a changing magnetic field and/or a loop moving in a non-uniform magnetic field, the emf induced in the loop is given by $$\mathcal E = -\frac{d\Phi_B}{dt} \iff \oint \mathbf E \cdot d\mathbf r = -\frac{d\Phi_B}{dt}.$$ The negative sign in Faraday's Law above indicates \highlight{Lenz's Law}: the emf induced in a loop of wire is such that the generated magnetic field opposes the change in the external magnetic flux.

Definition: Self-Inductance
A circuit element can induce a current within itself through its self-generated magnetic field. This is characterized by the inductance, which is defined as $$\Phi_B = LI, V = -\frac{d\Phi_B}{dt} =-L\frac{dI}{dt}.$$ The energy stored within an inductor is given by $$U = \frac 12 LI^2.$$

Definition: Mutual Inductance
If you have two inductors of inductance $L_1,L_2$, then the total magnetic flux through each is given by \begin{align*} \Phi_1 &= L_1I_1 +M_{12}I_2,
\Phi_2 &= L_2 I_2 + M_{21}I_1. \end{align*}

Theorem
The mutual inductances are always equal, $$M_{12} = M_{21}.$$

Idea: LR Circuits
If you connect a inductor with constant current to a resistor in series, its current will decay exponentially, $$I(t) = I_0 e^{-t/\tau} , \tau = L/R,$$ where $\tau$ is the time constant. Intuitively, the inductor always wants to keep the same amount of current flowing through it.