(1) Sudoku
This is the most famous of all the various logical puzzles, and is seen in virtually every newspaper each day, and also carried in many magazines. Because the rules are so simple to grasp and there is no maths involved, and various solving strategies can be used, this is a puzzle that looks set to stay.
(2) Kakuro
Kakuro is a puzzle that is a little like a number crossword in terms of the grid. There are sum clues given at the start of every set of cells, which is called a run, that may be horizontal or vertical. As with sudoku only 1 - 9 may be used, but here there is no repetition allowed within an individual sum rather than in the row or column at large. These puzzles can be quite tricky and a good quality maker is needed to make enjoyable kakuro for humans. As they involve maths they have not been as popular in the West by some stretch as sudoku, nor indeed as popular as they are in Japan.
(3) Shikaku
Shikaku requires a grid to be divided into rectangles of varying dimensions. At the start of the puzzle there are some cells with numbers in and those indicate that the square containing that number belongs to a rectangle of the size stated by that number. Each number must be in precisely one rectangle and it must be of the dimensions on that number. That's all there is to it: rectangles cannot overlap of course.
(4) Masyu
A lovely loop making puzzle but one that is rarely seen, this game requires a single continuous loop be drawn that does not have to include every cell in the grid. There are black and white counters which help you figure out how the loop moves: it must pass straight through a white counter but turn 90 degrees on one or both cells after. On a black square it must turn 90 degrees but then continue straight on for at least one cell either side.
(5) Slitherlink
The most well-known loop puzzle and also perhaps the hardest. In slitherlink a continuous loop must be drawn as above, but this time the grid is drawn with four dots at the corners of each cell, and most cells contain a number. This number tells you how many sides of that cell are touched by the loop. Thus the numbers range from 0 to 3, as clearly 4 would be a single closed loop around that cell and thus could only appear in the simplest of all slitherlinks.