What are lattice points exactly? For what purpose are they used in solid state chemistry?
There are different types of lattices. A lattice point is a point in any of these lattices. Lattices are used to describe highly ordered systems such as crystals and some supersolids. I am unsure whether lattices can only describe periodic systems (see quasicrystals). However, I will be assuming that we are talking about periodic lattices. I will also be assuming that we are dealing with crystals.
The crystal lattice is used to describe the lattice of a real crystal. For example, in NaCl, a lattice point in a crystal lattice represents the position of a sodium ion or a chloride ion.
Bravais lattices are more mathematical and abstract than crystal lattices. They are pretty much the same as crystal lattices. Unlike the crystal lattice, however, lattice points in the Bravais lattice no longer represent a position of a particular atom. Instead a lattice point represents a position in which an atom can be placed. In other words, a lattice point in a Bravais lattice is a point, which is equal and indistinguishable from any other another point. What matters in a Bravais lattice are not the points themselves, but how they are arranged (i.e. symmetry).
The reciprocal lattice is the Fourier transform of either the crystal lattice or a Bravais lattice. More often than not, it is used to refer to the Fourier transform of the Bravais lattice. More of that is explained in Physics.SE. However, I personally think that this video explains it better. A lattice point here can represent an atom or a point, depending on the context.
Finally, as mentioned by other posts, these points can be vacant.
How many Lattice Points are there in each of the Unit Cells?
Its depends on which type of Unit Cell we are talking about.
See this for full refrence of Unit Cells including their types.
lattice points are mathematical objects. In fact, a lattice is an infinite array of points in space where each point has identical surroundings to all others. A lattice is thus a purely abstract mathematical object.
In 3 dimensions there exist the 14 Bravais lattices filling all space. Their importance in solid state chemistry is that every crystal structure results from the convolution of one these Bravais lattice with a basis (a basis is an atom or a group of atoms).
Lattice points are the positions (or coordinates if you wish) where you can place an atom. Therefore you can find either atoms or vacancies in the lattice points of a crystal structure. I dont quite understand what you mean by "How many Lattice Points are there is each of the Unit Cells?"
The points in a crystal lattice, representing the positions of structural units(atoms,molecules or ions) in the representative crystal are called lattice points . Each lattice point in the crystal has the same environment as the structural units have in the real crystal. How many lattice point in the unit cell, depend on the type of crystal .