El centro simply means the downtown area, the main business district. ¿Me acompañas al centro? Wanna go downtown with me?
Spanish has a convenient word for saying that something is centrally located: céntrico. An apartment for rent that is advertised as céntrico won’t necessarily be right in the downtown area, but at least it won’t be on the outskirts of town.
When I first came to Mexico, I would get el centro confused with la central, short for la central camionera, the long-distance bus station. Don’t expect taxi drivers to decipher mistakes like *el central.
While we’re on the subject, a bus stop is a parada and the place where local buses wait between trips is a paradero, the end of the line.
A shopping mall is called a centro comercial or a plaza.