Futoshiki means inequality in Japanese, and that is the whole idea. Fill the grid with the digits 1 up to its size, once per row and column, exactly as in Sudoku. The difference is the signs between cells: a greater-than or less-than telling you which of two neighbours holds the higher number. The open side always faces the larger digit.
How to solve it
Look for chains. Three cells in a row with signs pointing the same way mean the values rise all the way along, and in a five-grid the first cell in such a chain can never exceed 3. The longer the chain, the tighter the constraint — one spanning the whole grid hands you every digit at once.
The extremes are your friends. A cell smaller than all its neighbours cannot hold the highest value, and a cell with signs pointing away from it on both sides is squeezed between a smaller and a larger digit. Always start where the signs cluster.
Combine that with ordinary row and column logic. The inequalities alone rarely finish a grid; it is the alternation between them and the uniqueness rule that cracks it.
What makes a puzzle hard
How many digits are given and how the signs are spread. Signs bunched in one corner constrain little, while long chains make a puzzle almost self-solving. Larger grids are harder too, because each inequality rules out a smaller share of the possibilities.
A worked example
In a five-grid you find three cells in sequence where the signs give a less than b less than c. That forces a to be at most 3, b to sit between 2 and 4, and c to be at least 3. No cell is decided yet, but a great many possibilities are gone — and a single given digit nearby is usually enough to topple the rest.
The most common mistake
Reading a sign backwards. The rule to remember is that the open, wide side faces the larger number and the point faces the smaller. A single misread sign early on makes the whole grid unsolvable, and the error is painful to track down afterwards.