Our moon is "gravtiationally locked" to the Earth (it's also sometimes called "tidal locking, but I think that phrase is potentially confusing). This means that only one side of the Moon is ever facing the Earth, but what would happen if the Earth was locked to the Sun?
Several things would happen. The obvious outcome being that one side of the Earth would gradually absorb the Sun's radiation, while the other side would get practically none. The entire planet would end up with one blisteringly hot side and one very cold one.
Where this gets interesting is if we consider life. Life on Earth needed liquid water to evolve. The chemicals we're made of have to be able to move around in order to transfer information and react with each other, that's why life is never found in water-less environments.
On our locked Earth, the only place liquid water could form would be on a thin band around the rim of the planet. On the "hot side" the water would evaporate, while on the other side all the water would be frozen as ice. No life would form on either of the two hemispheres. What's more, the hotter side of the Earth would also end up with lots and lots of heated air moving around, leading to crazy storms.
The life which did arise would be confined to a thin rim a few dozen kilometers wide stretching all the way around the planet. When life did arise there, it wouldn't have much room to spread out, meaning evolution would take longer to happen. As a result, the creatures which did manage to exist would be totally different from anything living today.