I am down here at the school. From the speaker's perspective, she is at a location which is geographically or metaphorically "down" from some other point. If I am at a store that is south of some geographic point (such as downtown or south of where I live), I can say I am down here at the store. You can also say this if you are talking to someone on the phone and you are south of where they are. "Hey, I am down (here) at the park." You can also use this expression if you are somewhere that is physically or geographically lower than a different reference point. "I am down (here) at the lake" means the lake is at a lower elevation than the person you are talking to over the phone. The same goes for "up here" and "over here," although "over here" has a more generic sense of just being away from some geographic reference point. That is what all these expressions mean: the speaker is referencing his location from some other location. This does not have to be a geographic point. "I'm down (here) at the farm" can either be geographic, if the farm is south of another point. Or it can be metaphorical, in that the speaker considers the farm to be "metaphorically" down ftom some other position or place.

As Su Ki mentioned, "up" and "down" can refer to altitude. We say "down at the beach" or "down at sea level" as Su Ki mentioned. We also say "up in the hills" or "up in the mountains". So, if someone has a cabin in the mountains, they might say "up at the cabin". With reference to rivers, "up" and "down" also refer to "upstream" and "downstream". We say "up in the stands", "up in the bleachers" or "up in the nosebleed seats" to refer to the highest and least expensive seats in a stadium. As Su Ki mentioned, "down" and "up" are not always physical in nature. One figurative meaning of "up" and "down" comes from New York City. In New York City, "uptown", "midtown" and "downtown" refer to northern Manhattan, central Manhattan and southern Manhattan. However, in American English at least, "uptown" and "downtown" have entered the language with connotations that transcend these geographical meanings. "uptown" suggests affluence and a more sophisticated and urbane culture. "downtown" suggests more hip and stylish in a more working class way. Billy Joel has a song that talks about the relationship between an "uptown girl" and a "downtown guy". New Yorkers also refer to "upstate" as being any part of New York State that is not New York City (because NYC is at the southernmost part of New York City). I don't know if any other state uses "upstate" in this way. "up" and "down" can also take hierarchical meanings. "up on the executive level" is used when the top managers have offices on the top floor of the corporate office building. "up at corporate" suggests that the corporate office building is figuratively higher than the other workers. At school, "upper classmen" are juniors and seniors (as opposed to freshmen and sophomores). "down at school" or "down at the hospital" are new to me. Can you provide some context about these usages?

'Down' sometimes refers to a place which is physically lower than another place, for example, it would be normal to say 'The fishermen were down at the beach', because you would expect the beach to be lower than most places. In other situations, 'down' simply means 'over here/over there' . It's used - especially in informal American English - to reinforce a contrast between one place and another. For example, you'd talk about a country boy who's made money working in the big city but 'meanwhile, down on the farm' his family are still struggling to earn a living. This doesn't literally mean 'down' - it's just the equivalent of 'back home.'

the second video is at https://www.youtube.com/watch?v=i_lzeHYNngE

For example, in the song "Down on the Bayou," those words can have both a literal meaning (at the bayou, which is _lower_ in elevation than the surrounding land); and a non-literal meaning (a popular place to gather for fun): https://www.youtube.com/watch?v=dB54Xhot_D8 There is also a song that is much slower and has the words, "Blue bayou," which you might like: https://www.youtube.com/watch?v=dB54Xhot_D8