I'm going to order the NE555 chip from Texas Instruments (Link) to build a circuit, but they only have NE555P, NE555DR etc. Can I use the NE555xx instead of NE555? What is the difference between them?
The difference is in the packages. The function is the same. You can search for package types on TI page, they have nice documentation. Datasheet
As you can see, in NE555
xx represent a different package type. There is more information about these
xx on pages 20-22 inside the datasheet. The letters NE/SA/NA/SE before the 555 represent slightly different manufacturing processes & part QA, corresponding mostly to safe operating temperature ranges (commercial, industrial, automotive, military) & voltages. You need to read the datasheet and select the chip which is appropriate for you.
Extra letters at the end of a part number usually specify the package, operating temperature range, quality level, or other such information. For the NE555, the list of part numbers is in chapter 12 of the datasheet in the Packaging Information table. For example, the NE555P is an 8-pin PDIP package rated for 0-70C. There's also information on lead finish, Pb-free/RoHS, package quantities, and moisture sensitivity level. Pick the parameters that matter to go and go with the cheapest part number that provides them.