All conures are parakeets, but not all parakeets are conures.
A parakeet can be anything from budgies and "grass parakeets" (bourkes, turquoisines, red rumps, scarlet chested keets), to rosellas, ringneck parakeets (psittacula) and even conures (aratinga, pyrrhura, nandayus, cyanoliseus, enicognathus, etc).
It's like saying that all macaws are parrots, but not all parrots are macaws.
Except for the Carolina Conure (most commonly called a Carolina Parakeet), parakeet is a general term used in the scientific community to define any small parrot with a long tapering tail.
This is not to be confused with budgies (aka budgerigar) that are often called "parakeets" in pet stores.
Conure is the term that is more readily used in the pet and aviculturist's community.
Are Conures their own type or are they a type of parakeet?
The correct answer would be "both".
Conures are also closely related to macaws. Aratinga essentially means "little macaw", as some of the macaw species are Ara.
Although conures are somewhat close to amazons as well, we have not discovered how closely related.
When I talk about relation, I'm talking about whether or not the two species can hybridize.