MS Access is a "relational" database like Oracle, SQL Server, DB2 and others, and one of the many features they all have in common is the ability to search through data using the "where like" function. This function is used by including "pattern match" and "wild card" characters in your criteria. They tell the search engine to get results without having an exact match - which is what it sounds like you want to do.
For example, try putting: John* in your query criteria. The asterisk is a pattern match character that tells Access to get every record where the first 4 characters are "John" and it doesn't matter what comes after.
The wild card is similar, but only works on a single character. So if you put: J??? Smith in your query criteria, it would return John, Josh, Jack and Jeff - as long as they all have the last name of Smith.
You can combine these characters into a single criteria - try different approaches so you can get a feel for how they work. Another example: Jo??*Smith* should return rows for John Smith and Josh Smithers, but not Jack or Jeff.
You can get more control by creating multiple criteria using "and" and "or" logic. To use "or" - enter more criteria on the row below (where it says "or"). Keep in mind that this returns rows where either criteria is true.
You can do "and" logic (where more than one criteria must be true) by adding the same column a second (third, fourth, etc) time to your query and putting more criteria on it. Note that in Access, you can keep the additional columns from appearing by un-checking the "Show" check box in the query.
If you want your search to ignore upper vs. lower case in your names, you can make the field all upper case by changing the way the field is selected. If you look at the "Field" row in your query definition, it normally just shows the field name.
Normally: NAME
Change to: Name: UCase([NAME])
If you add the Access function "ucase" it will convert the field to all upper case so that you can make your criteria all caps. This trick lets you match regardless of case.
One last thing - it might make sense to separate your name fields into: First name, Last name, Middle name, etc. This would make it easier to control how you create your searches since you could easily apply different logic to first name from what you apply to the last name.
Hope this long answer has some helpful info for you. The big thing is to just experiment & see how it all works. Have fun!