What is the name for the point in orbit closest to the Sun?

Perihelion, from the Greek for near the sun, with aphelion as its opposite. Every orbiting body has them, because orbits are ellipses rather than circles - and the Earth's perihelion falls in early January, which surprises people in the northern hemisphere who assume summer means being closer.

The naming changes with what you are orbiting. Around the Earth it is perigee and apogee; around a star generally, periastron; around a black hole, perimelasma. Kepler's second law is what makes the point matter: a planet sweeps equal areas in equal times, so it moves fastest at perihelion and slowest at aphelion, and the seasons are consequently not equal in length. Northern winter is a few days shorter than northern summer.

Mercury's perihelion also broke Newton. The whole ellipse rotates slowly, and it does so by a tiny amount more than Newtonian gravity predicts - so nineteenth-century astronomers proposed an undiscovered planet, Vulcan, and searched for it for decades. There was no planet. General relativity produced the observed figure exactly with nothing adjusted to fit, and Einstein wrote that it gave him palpitations.