The language you speak is English; the dialect is American English (or rather, American English is a group of dialects, one of which you speak). Similarly, British English is also a dialect of English, even though it can be thought of as the "original" dialect.
Dialects are defined precisely because languages vary in different regions, be they small or large. Just as I would say soda to refer to a carbonated beverage, others might say pop, coke, or one of many other terms. That doesn't mean we speak different languages; it only means that we use different words to refer to a specific concept.
At the heart of these dialects (or families of dialects), the core of the language is the same, even though the vocabulary may vary slightly. I can easily read a book in British English and don't have to "switch" my brain over to process the words differently—I may just have to remember the small variations in each language from time to time.
The question you may ask, however, is what separates a dialect from a language and when the former becomes the latter. To that, there is no definitive answer—only what is commonly accepted and understood both by the speakers of the dialect/language in question and by linguists who have studied the history of how they have evolved.
Wikipedia offers some influencing factors, especially noting the politics of the regions. This is true, but you should always keep this in mind: "How hard is it to understand a speaker of Dialect/Language B?" Etymological factors definitely make understanding related languages easier, but note the differences between Spanish and French (both Romance languages)—and then look at American and British English.