Hi Freesand,
You have Chlorophytum comosum aka spider plant.* You are correct that you see young plantlets. You can place them in a pot of soil while still attached to the parent.* When they establish with new roots you can cut the stem off.
http://www.hort.wisc.edu/mastergarde...piderplant.htm
They come in different named varieties, with some that are solid green and others that have different types of vareigation.
http://aggie-horticulture.tamu.edu/i...osum_Vitt.html
http://aggie-horticulture.tamu.edu/i...mosum_Var.html
http://aggie-horticulture.tamu.edu/i...m_comosum.html
Newt
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.