Resource: Forces overview
A general brainstorm for relevant and commonseen forces is listed below. Symbols are shown where often used and formulas where known and useful. (Be aware of vector versus magnitude notation, $\vec F$ and $F$, in formulas.) Symbols used:
 $\mu_k$ and $\mu_s$ are friction coefficients,
 $k$ the spring constant,
 $\Delta s$ elongated/compressed length (difference from natural/relaxed length),
 $C_D$ aerodynamic/drag coefficient,
 $\rho$ fluid density,
 $A$ frontal/projected area in direction of motion,
 $v$ speed and $\vec v$ velocity,
 $V$ submerged volume,
 $g$ gravitational acceleration,
 $m$ and $M$ two different masses,
 $G$ gravitational constant
 $r$ distance (orbit radius or charge separation)
 $q$ and $Q$ two different charges
 $\vec E$ electric field
 $\varepsilon_0$ vacuum permittivity
 $l$ length of wire carrying current $\vec I$
 $\vec B$ magnetic field
Mechanical forces (internal or needing contact)^{[1]}  

Tension  $$\vec T$$  ... in taut strings  
Compression  ... in lifting walls  
Normal force  $$\vec n$$  ... when a surface “holds back” against something  
Kinetic friction  $$\vec f_k$$  ... between sliding surfaces  $$f_k=\mu_k n$$  
Static friction  $$\vec f_s$$  ... between surfaces about to slide  $$f_s\leq \mu_s n$$  
Elastic force / spring force (Hooke's law)  $$\vec F_\text{elastic}$$  ... in stretched rubber bands or pressed springs  $$\vec F_\text{elastic}=k\;\Delta \vec s$$ 
Other usual forces such as kicking forces, throwing forces, chewing forces, etc. are included in those already listed.
Fluid forces (in liquids or gasses)  

Drag^{[1]}  $$\vec D$$  ... on flying planes and sailing submarines  $$D=C_D\rho Av^2$$  
Buoyancy^{[1]}  $$\vec B$$  ... pushing up the light bathing ring  $$B=\rho V g$$  
Diffusion^{[4]} ^{1}  ... causing absorption and mixing  
Surface tension / cohesion^{[2,3]}  ... causing water to form round rain drops 
Field forces (at a distance)^{[1]}  

Weight  $$\vec w$$  ... when gravity pulls you down  $$w=mg$$  
Gravitational force  $$\vec F_g$$  ... when gravity attracts  $$F_g=G\frac{mM}{r^2}$$  
Electric force (Coulomb's law)  $$\vec F_e$$  ... charge attraction and repulsion  $\displaystyle\vec F_e=q\vec E$ $\displaystyle F_e=\frac 1{4\pi\varepsilon_0}\frac{qQ}{r^2}$ 

Magnetic force  $$\vec F_B$$  ... magnetic pole attraction and repulsion  $\displaystyle\vec F_B=q\vec v\times \vec B$ $\displaystyle \vec F_B=l\vec I\times \vec B$ 

Strong nuclear force^{[7]}  ... holds the atom together  
Weak nuclear force^{[7]}  ... causes breakdown of atoms 
Weight and the gravitational force are in fact the same thing, we just must input $g=\frac{GM}{r^2}$.
The electric and magnetic forces are usually combined into one electromagnetic force (called the Lorentz’ force): $\vec F_L=q\vec E+q\vec v\times \vec B$.^{[1]}
Chemical forces  

Adhesion^{[11]}  ... when tape sticks and paint adheres  
Bonding force^{2}  ... holding materials together  
Capillary force^{[9]}  ... when liquid by itself sucks into a straw 
Many of the forces can be differently categorised, such as intermolecular forces (bonding force, strong nuclear force etc.).
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