Orthogonal means only going in the four cardinal directions. An orthogonal line will either be vertical or horizontal.
Diagonal can either refer to the 4 intermediate directions (up-right, up-left, down-right, down-left) or to any direction that isn't orthogonal. Assume it always refers to the former: only the four other compass directions.
Royagonal is an adjective I coined referring to the 8 compass directions: combining orthogonal and diagonal. Its name is based on the king and queen in chess, who both move royagonally.
Hippogonal means non-royagonal. For example, in chess, a knight's overall move direction (1 in one direction, 2 in another) is hippogonal.
An ordinary path consists of a single line made of individual orthogonal segments. The path never branches off, crosses itself, or forms loops. The segments may sometimes be straight lines between the centers of cells, or between dots on a grid, or something else.
An ordinary loop is an ordinary path whose start point and end point are the same. Loops are still disallowed, except for the one big loop the puzzle must form.
An ordinary tree is an ordinary path where branching is allowed, but looping is not. Self-crossing is still unallowed, so a four-way intersection must read as one "stem" with three "branches" to be legal.
A Hamiltonian path or loop is an ordinary path or loop that must cover every available space possible; for example, in a cell-based puzzle, it must cover every cell.
A margin number is a number right outside of a row or column that says something about that row or column. For example, a 1 on the left of a row might say "there's 1 pearl in this row" if the puzzle specifies as much.
Line of sight refers to whether or not a line can be drawn between two points in some direction without crossing some blocker. For example, two pearls don't have orthogonal line of sight if a cell between them is shaded (as long as a shaded cell is defined as blocking line of sight).
Directional contiguity specifies that a group of cells must be contiguous with respect to some direction. For example, "orthogonally contiguous" means that from any cell in the group, using only orthogonal steps, you can reach any other cell by crossing only other cells in the group. No member of the group is unreachable by being, for example, only diagonally connected to the rest of the group.