Let's say your cleric wants to cast a spell that helps his companions and the radius of the spell is a 20 foot radius centered on him. That's great, now we just need to see who is within 20' feet. Take a look at this picture:
Let's use one of the diagrams from the book to help us out. The next picture includes the diagram and clearly shows which characters are in or out of range. But you might also notice something else about the picture and that is that the 20' radius of the spell cast by the cleric is not centered on the cleric. It extends 20' to his right, but only 15' to his left. This is our dilemma to solve.
We have a bit of a problem. Any radius centered on a person, in the middle of a square, is going to extend midway through the squares at the edge of the circled area encompassed by the spell.
So we could truly center the spell on the caster, so that it does not align with the grid and in doing so, the spell would partially touch every character in the picture as shown here, even though it does not line up with the base grid below it.
What to do? We came up with a solution that we feel is best because it is easy to implement, and gives the caster a good amount of control over the result. It is also based on the rules from pages 28 of the DMG: "To employ the spell using a grid, the caster needs to designate an intersection of two lines on the grid as the center of the effect. From that intersection, it’s easy to measure a radius using the scale on the grid. "
We allow the caster to pick one of the intersections of one of the four corners the caster is standing in, and that corner becomes the center of the 20' burst allowing us to use the diagrams from the book, as in the second picture above, to determine those affected by the spell.