"Ha usato una mappa"
is the main/independent clause;
"per trovare il paese"
is a subordinate/dependent implicit final clause.
The implicit final clause usually requires
per + infinitive mood of the verb
If the main clause has a verb expressing movement, you can use
a + infinitive mood of the verb
"Si è messo a cercare il paese sulla mappa"
In some cases you can also use "allo scopo di", "al fine di", "onde", etc. (these are more formal, more common in writing).
Note that the subject of an implicit final clause is the same subject of the main clause; if this is not the case you need to use an explicit final clause (affinché, perché, etc.). Example: "Lui tirò fuori la mappa affinché/perché lei cercasse il paese".
Also note that in this answer I am assuming that the question is about final clauses - although there are obviously many other cases where you use the infinitive of a verb, and it would be too long to describe each case here.