Start with whites. White goes with everything, though bright whites can be too intense for warm tile colors, off-whites are less edgy. In terms of matching, you can't go wrong with any off-white or white, no matter what the tile color.
Try colors in the same color range as the tile. For instance, with a terra cotta tile, a warm clay red-orange, try similar red-orange in lighter or darker shades. Try the same for other colors, like gray-green tiles. For black or white, try grays, though this may be too tame, and you may wish for something more contrasting.
Match tiles against colors further away on the color wheel but with the same color temperature. The terra cotta in this example would go well with maroon, on the red side, or a warm yellow. Try different swatches as examples, placed against the tile. Do not pick a color too close to the tile color, as that can cause unpleasing clashing. For black, gray or white tiles, you may select virtually any color to match with the tiles. Look to existing furnishings to match for ideas.
Use complementary colors for strong contrast. For red, select green; for yellow, select purple; for orange, select blue. Hence, for the terra cotta, a reddish tone, look in the green range. For an orange-brown tone, try a Mediterranean blue. These are bolder choices, not for all tastes.