The Short Version: A Blue Moon Is a Calendar Effect
A full moon occurs when the Moon is near the opposite side of Earth from the Sun, so the side facing us is almost fully illuminated. A repeating lunar phase cycle is called a synodic month, averaging about 29.5 days. Because calendar months are often 30 or 31 days long, one month can occasionally contain two full moons. The second one is commonly called a Blue Moon.
Core picture: Blue Moon is not a color change on the lunar surface. It is a timing effect caused by a 29.5-day lunar phase cycle fitting inside a longer calendar month.
Why Does a Full Moon Happen?
The Moon does not shine by itself; moonlight is reflected sunlight. When Earth lies roughly between the Sun and the Moon, the Moon's Earth-facing side is nearly fully lit. The Moon's orbital tilt is why most full moons do not become lunar eclipses.
Why Is the Synodic Month About 29.5 Days?
The Moon orbits Earth relative to the stars in about 27.3 days, but Earth also moves around the Sun. To return to the same Sun-Earth-Moon geometry, the Moon has to travel a little farther. That makes the phase cycle about 29.5 days.
This is why the lunar phase cycle can drift against the human calendar and occasionally create two full moons in one month.
Does a Blue Moon Change Tides?
A Blue Moon is just a full moon with a special calendar label, so it does not add extra gravity. Tides depend on the Sun-Earth-Moon alignment and the Moon's distance. Full moons and new moons can produce stronger spring tides, but the word "blue" adds no new physical effect.