It is spider plant, Chlorophytum, probably C. comosum.
It is normally seen as a variegated cultivar, but yours is almost plain green.
Yes, you can cut off offshoots and pot them up separately. I would trim back the leaves by half at the same time.
You can also divide the whole plant. After a while, it will fill the pot with its fat white roots (you'll wonder where all the soil went!)
It should also produce flowers on stems, which lengthen, arch over and touch the ground, where they will root. Roots and new leaves often start forming before reaching the ground. You can cut off these little plants and pot them up.
Thanks, Mike! I think you are spot on.
It might be because in my country Belgium, even in summer it's never much hotter than 28°C or because it hangs here between shadow and direct window light, but I never had flowers in 3 years. Though I love the plant even without them! Hangs great in middle of room!
Thanks again, dude!
About to split them soon (I like "the spiders" too much). :-)
P.S. It has some whit stripes and has been whiter before. Dunna why it's that green. Still love it. :-)