See imgur link for pics of it leaning: https://imgur.com/a/M6rKe

It is a Dracaena marginata. It looks a little worse for the ware. It looks like overwatering damage with some rot probably starting, which would leave stems weak enough to lean over. Look at the trunk down near the soil. It is discolored dark right out of the soil, then there looks like almost a waterline where it gets whitish, and then looks a bit wrinkled, above that. Does that section feel firm and normal? The leaves are yellowing, the way mine did when I overwatered it.

Does the pot have holes for draining and allowing salts to be leached out of the soil. Years of watering a big plant with nowhere for water to ever run out is lethal sooner or later. If it were mine, I would have a look at the roots, to see if they are firm and white or are they dark, withered, and rotting. Does the soil smell like potting soil or does it stink. Whatever the answers, you will be able to save the tops and reroot them to make a nice new plant. If you feel the trunks have nothing wrong and are just leaning from too much growth, then they will sprout new tops for you too. There are sites you can google for great instructions.

The plant you showed a picture of is a yucca. It will want less watering, and more sun. Keep us posted and Good luck.

Just by the way, my dracaena marginata's leaves drooped like yours until it went out on the porch this summer, and all the leaves that grew since then are all spiky.

More sunlight would also help strengthen and thicken the stem, though it might need support to keep it from bending over until it could stay upright by itself.

I look forward to seeing more info about your plant, Christina.

I had one I worried about that perked up well, when I lowered watering and increased light. I hope this is all Christina needs too.

Thanks to all for the replies. I apologize for being MIA - I've had a busy few weeks and haven't been able to devote time to a decent reply. Thank you for your patience :)

"Look at the trunk down near the soil. It is discolored dark right out of the soil, then there looks like almost a waterline where it gets whitish, and then looks a bit wrinkled, above that. Does that section feel firm and normal?"
The whole trunk feels firm and normal, "waterline" discoloration included. The "waterline" line has been there since about the 1st year.

"Does the pot have holes for draining and allowing salts to be leached out of the soil. Years of watering a big plant with nowhere for water to ever run out is lethal sooner or later."
The pot does not have holes for draining. I plan to pick up a new pot this weekend - hopefully the local Home Depot or Lowes sells clay pots with holes in the bottom, as I do like the look :) - I'm assuming I should shake out as much of the soil as possible and replace it. But with what? Just a generic potting mix? When I first got it I potted it in this container with Miracle Gro (which I now know is bad!) and it has never been re-potted.

Shortly after writing my original post I moved the plant into the den/2nd bedroom. It's the only room in the apartment that gets direct sunlight - it gets several hours of afternoon sun. All yellow leaves have since dropped but I am left with several that are dry at the ends. I haven't watered it since the day I wrote my original post and the soil still does not yet feel excessively dry.

Do you still have your plant? this was a few years ago but I have one too. If it was in the corner then it was leaning for light. You can prune them as well. Google how to prune my dracaena margintina. Curious how you made out.