"Backward" has negative connotations here, in the west. But generally, the abbreviation "OBC" (Other Backward Castes) is used. From my personal observation, SC/ST/OBC folks are as caste-conscious as anyone else, but have successfully translated their caste identity into political power in India. In fact, it is fair to say that people from these categories (who comprise a numerically significant portion of the population) have a pretty good grip on power at every level of government, down from your local municipality ward up to the offices of the Prime Minister/President (do you know that the current Prime Minister of India is from a OBC)?
Generally, in polite conversation, no one ever talks about the other person's caste. A lot of the time, this is unnecessary, though. There are tons of caste signifiers, such as your last name, your skin tone, your language (both speakers can use the same language but the set of words used and usage/abusage can tell you a lot about the person's background. Like how you could tell a cockney from an upper-crust West Londoner.) Don't even get me started on food. If you are a "pure vegetarian", then it's a very good chance you are high-caste. With the professions also, things are changing a bit more quickly, but age-old divisions remain. You would be hard-pressed to find a Brahmin working a construction site (there's a good chance a Brahmin would be in an air-conditioned office drawing up the plans for said site); almost zero chance you'd see an upper-caste person going out fishing to sell the catch and make a living, and so on.
If you participate in a conversation with a group of Indians who have just met (at the workplace, say), just observe how they interact: they will put out direct or subtle questions like: "Where are you from originally?", "Where did you attend college?", "What do your parents do?", "Are you vegetarian or non-vegetarian?"... It just goes on and on. You can tell a lot about a person within 15 minutes of meeting him, if you have been raised in India.
EDIT: BTW, the terms "Scheduled Caste", "Scheduled Tribe" (perhaps even "Other Backward Classes") were introduced by the British! The British bureaucrats who ruled the country had an obsession with making tidy lists to categorise people (hence the "Schedule" in "Scheduled"!) and came up with the (original) lists of these castes and tribes!