Definition: Biot-Savart Law
The Biot-Savart Law allows us to find the magnetic field $\mathbf B$ at a given point in space via $$d\mathbf B=\frac{\mu_0}{4\pi}\frac{I\, d\mathbf l\times \hat {\mathbf r}}{r^2}$$where $\hat{\mathbf r}$ is the direction vector from the current source $I\, d\mathbf l$ to the point where we are finding the magnetic field.

Definition: Ampere's Law
For symmetric current distributions, we can find the magnetic field easily using Ampere's Law, which states that the path integral of $\mathbf B$ along a closed loop is proportional to the current passing through the loop: $$\oint \mathbf B\cdot d\mathbf s=\mu_0I_{enc}$$The positive direction of $I_{enc}$ is defined via the right hand rule.
Using Ampere's Law, one can find the magnetic fields due to some standard setups, i.e. from an infinite line of current ($B=\mu_0I/2\pi r$) and inside an infinite solenoid ($B=\mu_0nI$).

Definition: Gauss's Law for $\mathbf B$-fields
There do not exist magnetic ``point charges" (magnetic monopoles), so Gauss's Law applied to magnetism states that the net magnetic flux through a closed surface is always 0: $$\oint\mathbf B\cdot d\mathbf A=0$$Consequently, magnetic field lines must always trace back in a closed loop, and cannot begin or end at magnetic monopoles.

Definition: Lorentz Force
The force acting on a charged particle $q$ in an electric field $\mathbf E$ and magnetic field $\mathbf B$ is $$\mathbf F=q(\mathbf E+\mathbf v\times \mathbf B)$$Take note that a magnetic field cannot do any work on a particle.

Consequently, the force acting on a current due to a magnetic field is given by $$d\mathbf F=I\, d\mathbf l\times \mathbf B$$

Definition: Magnetic Dipoles
A magnetic dipole is any closed loop of current. The dipole moment $m$ of a loop of area $A$ and current $I$ is defined as $$m=IA$$ The magnetic field due to a magnetic dipole is equal to (just like with electric dipoles) $$\mathbf B=\frac{\mu_0m}{4\pi r^3}(2\cos\theta\, \hat{\mathbf r}+\sin\theta\,\hat{\mathbf \theta})$$

Tip: Modeling Dipoles with Monopoles
At large distances away, the magnetic dipole behaves identically to its electric counterpart, such that we can assume it to be composed of two magnetic monopoles of charge $\pm q_m$ separated by a distance $d$ such that $q_md=m$. You can use all formulas associated with electric dipoles to solve problems with magnetic dipoles, so long as you remember to use $\mu_0/4\pi$ for magnetism instead of $1/4\pi\epsilon_0$ for electricity.

Definition: Magnetic Field Energy Density
This idea was not covered in lecture, but the energy density associated with a magnetic field is $$\frac1{2\mu_0}B^2$$You can use this in the exact same way as the electric field energy density.