I believe you're looking for number (grammatical number), as in singular vs plural. The verb must agree in number with its subject.
In the third person ("the car"), the form does is singular, and do is plural. (So you'd say "The car doesn't run", using the singular does because car is singular, but "The cars don't run", using the plural do because cars is plural.)
+--------------+--------------------------+--------------------------+
| | Singular | Plural |
| | | |
+--------------+--------------------------+--------------------------+
|Third person | does | do |
| | The car doesn't run | The cars don't run |
+--------------+--------------------------+--------------------------+
|Second person | do | do |
| | You don't run | You (all) don't run |
+--------------+--------------------------+--------------------------+
|First person | do | do |
| | I don't run | We don't run |
+--------------+--------------------------+--------------------------+
In modern English, with most verbs, it seems that the third person plural form is the same as the one used for both second and first person, in both singular and plural: only the third person singular is different.
(An exception is "be", which in the first person singular is "am".)
In many other languages, all forms are usually distinct. English also had e.g. the form like "dost" that was used for (first and) second person singular ("thou dost"), which is now lost.