Your plant is a Yucca elephantipes that is near death, maybe someone else can help you for I know little about them..........

This plant you have rescued may have received poor care before it was abandoned. If you haven't already, spread out some newspapers and carefully take the plant out of its pot to check the roots. If you find lots of firm white or light brown roots, congratulations. If you find smelly rotten roots, the plant may be beyond help. (I say "may" because saving it then will require cutting the stem off and inducing it to grow new roots. Please ask for more help if you think this is what it needs. Chopping a plant down can be scary.)

If the roots are fine, I recommend you look up the plant using the Latin name provided by TeenGardener1888, and then base your care on what the plant needs. Every day, gently stick your finger into the potting mix, feel whether it is damp or dry, and then decide whether to water. You may find it needs water every three days, or every two weeks, instead of weekly. Once it recovers and begins active growth, the yucca's water needs may change, so keep checking.

Also, I often move plants to take photos. Is the yucca in your photo living in that spot, or there for the photo? If that's home for the yucca, you may need to move it to a brighter spot. I can't tell from the photo how much sun the plant gets.

Thank you for your responses! It wasn't in much better shape when I got it, the leaves were a bit greener though on the taller portion. I haven't checked the roots, but will. The spot it is in in the photo, is where I have been keeping it. It is in our kitchen which has a sliding glass door with no curtain so it's pretty bright in there. Sun comes in through that door in the afternoon and the plant is in the direct sunlight at that time. Should I move it closer to a large window?

I would put it as close to the window as you can. In most homes the quality of light diminishes quickly as you get a few feet from windows.

Overwatering might be an issue causing the leaves to yellow and roots to rot. It is recommended that the soil dry out between waterings.

If Teen hadn't said Yucca, I'd have ID'd your plant a Dracaena.
I've never seen a U-shaped side branch on a Yucca before.

The trunk looks firm. Is it?
How warm is the kitchen?

Yucca is succulent, therefore soil should dry between watering.
Required brightest/direct sun. Place closer to a window.

Good luck. Toni

I will move it next to a window. Our house is usually cool - kept aroun 64 this time of year. The trunk is very firm but the leaves are still dying.

Your plant won't need much water during winter, especially in lower temps so make sure it is dry before the next watering.