a lattice plane
A crystal lattice is an endless three-dimensional grid of points, like the corners of stacked identical boxes filling all of space. Slice through this grid with a flat sheet, and if your slice passes through a regular two-dimensional array of the grid points, it is a lattice plane. The lattice is completely filled by stacks of parallel, equally spaced lattice planes — think of the same grid viewed as a deck of evenly spaced cards, and there are many ways to deal the deck.
Each family of parallel lattice planes is labelled by its Miller indices (hkl), and the indices have a direct geometric meaning: the planes cut the cell edges a, b and c into h, k and l equal parts respectively. Every lattice point belongs to some plane of the family, the planes are separated by the interplanar spacing d_hkl, and these are precisely the planes that reflect X-rays in Bragg's law. Choosing larger indices deals the deck into more, thinner slices with a smaller spacing.
One warning captures the deepest idea in crystallography: lattice planes are planes of the lattice, which is pure geometry, while the actual atoms — the motif — merely decorate those planes. A sheet of atoms and a lattice plane coincide only in the simplest structures, where a single atom sits at each lattice point. Confusing the lattice (where you stamp) with the crystal (the stamp plus its picture) is the classic beginner's error, and it hides here too: what diffracts is set by the lattice planes, but how strongly it diffracts is set by the atoms decorating them.
The (111) lattice planes of a face-centred cubic lattice slice each cell edge into one part and stack perpendicular to the body diagonal, spaced d = a/sqrt(3) apart. Bragg's law uses this spacing to place the (111) diffraction peak, but the peak's intensity comes from the atoms sitting on the planes.
Lattice planes are sheets of the lattice, labelled (hkl); the motif decorates them.
A lattice plane is a plane of the lattice (geometry), not necessarily a plane of atoms. The two coincide only when one atom sits at each lattice point; otherwise the motif decorates the lattice planes.