Lattice points are the positions an atom, ion or a molecule can occupy in a crystal giving it its shape and characteristics. If any of these particles go missing or are not in a correctly ordered arrangement, it can introduce a defect in the crystal. Lattice points are essentially a crystalline framework.
For example in an ice crystal, the lattice points would be the locations where the water molecules can be found. These water molecules are held in place by the Hydrogen bonding that gives the ice crystal its hexagonal ring structure.
As the crystalline solid is a collection of multiple symmetric periodically repeating unit cells, the lattice points that belong to one unit cell can only be found out if the contribution that each atom makes towards a unit cell is calculated.
For example, in a simple cubic unit cell, atoms or the lattice points are present at the corners of the cube. But each corner of a cube is shared with eight other unit cells- four from the front and back. So the net contribution of one atom present at the corner to one unit cell is only 1/8th. As there are eight corners in one cube and 1/8th of an atom in a corner, so the total number of atoms per cube/unit cell is one. (1/8 x 8 = 1)
In other words, the number of atoms/lattice points that belongs to one simple cubic unit cell is 1.
Similarly, for a Body-centered cubic unit cell, in addition to the corners, lattice point is present at the center and is not shared with any other unit cell. So the total number of lattice points in a body-centered cubic unit cell is 2. (1/8 x 8 corner atom + 1 center atom)
In a face-centered unit cell, there are six faces. Each face has half of an atom that is present in one unit cell, and the rest is a part of the adjacent cell. So, the total number of lattice points that belongs to one unit cell is 4. (6 faces x 1/2 atom at each face + 1/8 x 8 corner atom)
For an edge centered unit cell having 12 edges, an atom at the border is shared with four unit cells- two at the top and bottom. Its contribution is only 1/4th to a unit cell. The number of lattice points per unit cell is therefore, 4. (12 edges x 1/4 atom at each edge + 1/8 x 8 corner atom)
Please note that for simplicity, here we have imagined an atom, ion or molecule as round spheres.