Begin by removing everything in the sink and taking out the items from the cabinet beneath, as sometimes leaks and other issues may cause damage below. Place a bucket beneath the drain pipe to catch water and debris, and make sure that it is large enough to cover the area to avoid spills and debris damage. Remove the grate by loosening the slip nuts at the appropriate sections around the grate and drain.
The slip nuts are designed to hold the tailpiece in place. When they are removed, pull the tailpiece apart, and set it away from the assembly. There may be debris or water left over in the tailpiece, so allow it to spill into the bucket. There is a strainer nut to undo, so remove the rubber gasket as well, as this is the last piece keeping the assembly together. When all of the items have been removed, simply push the drain hole up.
To replace, simply set all of the components back, and set the new grate in place of the old one. Make sure that all of the nuts are secure, and test the connection by running some water from the faucet to see if there are any errors in the connections.