Dungeons & Dragons nowadays is often played with some sort of grid, whether it’s a quick sketch on graph paper or a complete tabletop with miniatures. The game assumes a square grid, but there are other options with varying popularity.
You can print out graph and hex paper at Incompetech’s graph paper website.
Square is by the most popular grid layout, and is assumed in both third and fourth edition D&D rules. A point of contention between fans of these editions is diagonal movement - 3e charges two squares of movement for every second diagonal you move, whereas 4e takes the simpler and less realistic option of diagonal movement.
Square grids make it easy to draw rectangular rooms. It’s also a lot easier to draw your own grid on something like a whiteboard. Because it’s standard, most game books and products are made for square grid, and for this reason it’s my preference.
The main advantage of the hexagonal grid is in diagonal movement - there isn’t any. Straight lines are extended as a shortest path, of which you can have multiple due to the nature of a hex grid. Area effects are also easily adjudicated.
Third edition’s Unearthed Arcana gives rules for using a hex grid, which demonstrate how easily area effects are handled. For a 20ft radius, one “origin” hex and any hex within four hexes of this is affected. For a cone, it’s the same but constrained to two sides of the hex. Facing is more complex.
In fourth edition, hexes have no official support but can work reasonably well. Bursts affect less squares overall as they go from cubes to something more spherical, but six sides instead of eight mean fewer attackers can swarm you. A two-man flank is just as easy to perform. This might be an option if you’re unhappy with 4e’s simpler diagonal movement.
A method favoured by tactical wargames, this is the hardest to adjudicate. It requires the DM to measure the distances between units, decide whether or not opponents are “adjacent” or “in range”, and measure movement using measuring tapes or string. Flanking is likewise difficult to measure. This is the most flexible, but also the hardest to adjudicate.