Our neighbor has the tree in the picture, and he offered to let me cut a section of it so that I can regrow it in my yard.
Where should I cut? Or does it have seeds that I can replant?
This plant is a Bougainvillea. They flower reliably and grow in a wide variety of soils, like a lot of sun and it is easy to root cuttings provided you can give plenty of bottom heat. There is a previous discussion on this subject here which contains notes from others who have experience propagating the cuttings.
A couple of things to note: first this is not a tree per se, it is a vine, so it will nearly always be seen in association with a fence, wall, post or other support, even producing multiple stems so they can support each other after a fashion. Second it has quite vicious thorns, so watch carefully when taking the cuttings, wear heavy gloves - you have been warned.