Creating a Clickable Grid in a Web Browser

First, I encourage you to read this answer to another question involving the HTML5 Canvas. You need to understand that there are no squares. In order to detect a click on a 'square', you would have to keep track of a mapping from each canvas coordinate to the square(s) that it logically contains, handle a single click event on the entire canvas, work out which square(s) you want to change, and then redraw the canvas with the changes you want.

Then—since you seem to have no objection to using a more appropriate technology—I encourage you to do this in either HTML (where each 'square' is something like a <div> that is absolutely-positioned and sized and colored using CSS), or SVG (using <rect> if you need the squares to be able to be rotated, or want to introduce other shapes).

HTML and SVG are both 'retained-mode' graphics mode systems, where drawing a shape 'retains' the concept of that shape. You can move the shape, change its colors, size, etc. and the computer will automatically redraw it for you. Moreover, and more importantly for your use case, you can (with both HTML and SVG):

function changeColor(evt){
  var clickedOn = evt.target;
  // for HTML
  clickedOn.style.backgroundColor = '#f00';

  // for SVG
  clickedOn.setAttribute('fill','red');
}
mySquare.addEventListener('click',changeColor,false);

Edit: I've created a simple implementation in JavaScript and HTML: http://jsfiddle.net/6qkdP/2/

Here's the core code, in case JSFiddle is down:

function clickableGrid( rows, cols, callback ){
  var i=0;
  var grid = document.createElement('table');
  grid.className = 'grid';
  for (var r=0;r<rows;++r){
    var tr = grid.appendChild(document.createElement('tr'));
    for (var c=0;c<cols;++c){
      var cell = tr.appendChild(document.createElement('td'));
      cell.innerHTML = ++i;
      cell.addEventListener('click',(function(el,r,c,i){
        return function(){ callback(el,r,c,i); }
       })(cell,r,c,i),false);
    }
  }
  return grid;
}

As the accepted answer shows, doing this in HTML/CSS is easiest if this is all your design amounts to, but here's an example using canvas as an alternative for folks whose use case might make more sense in canvas (and to juxtapose against HTML/CSS).

The first step of the problem boils down to figuring out where in the canvas the user's mouse is, and that requires knowing the offset of the canvas element. This is the same as finding the mouse position in an element, so there's really nothing unique to canvas here in this respect. I'm using event.offsetX/Y to do this.

Drawing a grid on canvas amounts to a nested loop for rows and columns. Use a tileSize variable to control the step amount. Basic math lets you figure out which tile (coordinates and/or cell number) your mouse is in based on the width and height and row and column values. Use context.fill... methods to write text and draw squares. I've kept everything 0-indexed for sanity, but you can normalize this as a final step before display (don't mix 1-indexing in your logic, though).

Finally, add event listeners to the canvas element to detect mouse actions which will trigger re-computations of the mouse position and selected tile and re-renders of the canvas. I attached most of the logic to mousemove because it's easier to visualize, but the same code applies to click events if you choose.

Keep in mind that the below approach is not particularly performance-conscious; I only re-render when the cursor moves between cells, but partial re-drawing or moving an overlay to indicate the highlighted element would be faster (if available). There are a lot of micro-optimizations I've ignored. Consider this a proof-of-concept.

const drawGrid = (canvas, ctx, tileSize, highlightNum) => {
  for (let y = 0; y < canvas.width / tileSize; y++) {
    for (let x = 0; x < canvas.height / tileSize; x++) {
      const parity = (x + y) % 2;
      const tileNum = x + canvas.width / tileSize * y;
      const xx = x * tileSize;
      const yy = y * tileSize;

      if (tileNum === highlightNum) {
        ctx.fillStyle = "#f0f";
      }
      else {
        ctx.fillStyle = parity ? "#555" : "#ddd";
      }
      
      ctx.fillRect(xx, yy, tileSize, tileSize);
      ctx.fillStyle = parity ? "#fff" : "#000";
      ctx.fillText(tileNum, xx, yy);
    }
  }
};

const size = 10;
const canvas = document.createElement("canvas");
canvas.width = canvas.height = 200;
const ctx = canvas.getContext("2d");
ctx.font = "11px courier";
ctx.textBaseline = "top";
const tileSize = canvas.width / size;
const status = document.createElement("pre");
let lastTile = -1;

drawGrid(canvas, ctx, tileSize);
document.body.style.display = "flex";
document.body.style.alignItems = "flex-start";
document.body.appendChild(canvas);
document.body.appendChild(status);

canvas.addEventListener("mousemove", evt => {
  event.target.style.cursor = "pointer";
  const tileX = ~~(evt.offsetX / tileSize);
  const tileY = ~~(evt.offsetY / tileSize);
  const tileNum = tileX + canvas.width / tileSize * tileY;
  
  if (tileNum !== lastTile) {
    lastTile = tileNum;
    ctx.clearRect(0, 0, canvas.width, canvas.height);
    drawGrid(canvas, ctx, tileSize, tileNum);
  }
  
  status.innerText = `  mouse coords: {${evt.offsetX}, ${evt.offsetX}}
  tile coords : {${tileX}, ${tileY}}
  tile number : ${tileNum}`;
});

canvas.addEventListener("click", event => {
  status.innerText += "\n  [clicked]";
});

canvas.addEventListener("mouseout", event => {
  drawGrid(canvas, ctx, tileSize);
  status.innerText = "";
  lastTile = -1;
});

EDIT: Using HTML elements rather than drawing these things on a canvas or using SVG is another option and quite possibly preferable.

Following up on Phrogz's suggestions, see here for an SVG implementation:

jsfiddle example

document.createSvg = function(tagName) {
    var svgNS = "http://www.w3.org/2000/svg";
    return this.createElementNS(svgNS, tagName);
};

var numberPerSide = 20;
var size = 10;
var pixelsPerSide = 400;



var grid = function(numberPerSide, size, pixelsPerSide, colors) {
    var svg = document.createSvg("svg");
    svg.setAttribute("width", pixelsPerSide);
    svg.setAttribute("height", pixelsPerSide);
    svg.setAttribute("viewBox", [0, 0, numberPerSide * size, numberPerSide * size].join(" "));

    for(var i = 0; i < numberPerSide; i++) {
        for(var j = 0; j < numberPerSide; j++) {
          var color1 = colors[(i+j) % colors.length];
          var color2 = colors[(i+j+1) % colors.length];  
          var g = document.createSvg("g");
          g.setAttribute("transform", ["translate(", i*size, ",", j*size, ")"].join(""));
          var number = numberPerSide * i + j;
          var box = document.createSvg("rect");
          box.setAttribute("width", size);
          box.setAttribute("height", size);
          box.setAttribute("fill", color1);
          box.setAttribute("id", "b" + number); 
          g.appendChild(box);
          var text = document.createSvg("text");
          text.appendChild(document.createTextNode(i * numberPerSide + j));
          text.setAttribute("fill", color2);
          text.setAttribute("font-size", 6);
          text.setAttribute("x", 0);
          text.setAttribute("y", size/2);
          text.setAttribute("id", "t" + number);
          g.appendChild(text);
          svg.appendChild(g);
        }  
    }
    svg.addEventListener(
        "click",
        function(e){
            var id = e.target.id;
            if(id)
                alert(id.substring(1));
        },
        false);
    return svg;
};

var container = document.getElementById("container");
container.appendChild(grid(5, 10, 200, ["red", "white"]));
container.appendChild(grid(3, 10, 200, ["white", "black", "yellow"]));
container.appendChild(grid(7, 10, 200, ["blue", "magenta", "cyan", "cornflowerblue"]));
container.appendChild(grid(2, 8, 200, ["turquoise", "gold"]));