Replacement grips are the ones that come with the racquet when you originally purchase it.It has an adhesive on the back and goes directly on top of the wood part of the racquet.
They come in differnt types soft PU,leather etc.They are heavier,thicker and last longer than the overgrips.
Things to look out for when comparing replacement grips are cushiness or softness,durability,sweat absorption and tackiness.Higher the tackiness better the grip on the racquet but more friction with the hands.
They need to be replaced as soon as signs or wear become evident.

Overgrips are the ones that are supposed to be put on top of the replacement grip.They are a lot thinner than a replacement grip and dont have an adhesive on the back.They also weigh a lot less and dont last as long as the replacement grips.The same points have to be looked out for as for replacement grips.
They need to be changed more often as soon as signs of wear become evident.

It is highly advisable to change the factory grip as soon as it is purchased as they are normally of an inferior quality.Playing with a bad quality grip or a grip thats either too big or too small can lead to injuries.So its best to experiment with the exact fit.

It is best to experiment with a lot of different combinations i.e 1 replacement grip+1,2 or 3 overgrips to find out which suits you best.Men with small hands and women generally prefer just the replacement grip, while those with big hands like to bulk up their grips.

Please note that this article is written with mostly badminton in mind but also applies to squash and tennis to an extent.
Good luck and happy shopping!!

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