Friction
Friction is a force that resists the sliding or rolling of one surface over another and always acts opposite to the motion.
Definition
Friction is a force that pushes against things when they rub together; it slows moving objects down and makes it harder to slide one thing across another. It is a contact force that opposes the relative motion, or attempted motion, of surfaces in contact: static friction prevents an object from starting to move, kinetic (sliding) friction acts on a moving object, and both depend on the types of surfaces and how hard they are pressed together, converting kinetic energy into thermal energy. Kinetic friction has magnitude $f_k = \mu_k N$, where $N$ is the normal force and $\mu_k$ is the coefficient of kinetic friction for the surface pair, while static friction adjusts to whatever is needed up to a maximum, $f_s \leq \mu_s N$, with $\mu_s > \mu_k$ typically. Friction is independent of contact area to a good approximation and arises from electromagnetic interactions between surface irregularities. It is non-conservative: the work it does depends on path length and is dissipated as heat.
Example
Sliding across a gym floor in socks is easy, while sliding in sneakers is hard, because rubber shoes have more friction with the floor. A $4\ \text{kg}$ box pushed with $20\ \text{N}$ that only accelerates at $2\ \text{m/s}^2$ has a net force of $4 \times 2 = 8\ \text{N}$, so friction must be pulling back with $20 - 8 = 12\ \text{N}$. For a $10\ \text{kg}$ crate on a floor with $\mu_s = 0.50$ and $\mu_k = 0.30$, the normal force is $98\ \text{N}$, so it takes more than $0.50 \times 98 = 49\ \text{N}$ to start it moving. Once sliding, friction drops to $0.30 \times 98 = 29.4\ \text{N}$, and a steady $49\ \text{N}$ push then produces $a = (49 - 29.4)/10 = 1.96\ \text{m/s}^2$.
Key Insight
Friction is helpful more often than it is annoying; without it you could not walk, hold a pencil, or stop a bike. Static friction is usually stronger than kinetic friction, which is why it takes a bigger shove to get a heavy box moving than to keep it sliding. Antilock brakes exist for exactly this reason. A rolling tire grips the road with static friction, while a skidding tire slides with weaker kinetic friction, so a car that locks its wheels takes longer to stop.