I'm working at an office in Texas and there is an old tree with this mossy fuzzy stuff. It looks like it's leeching off the tree. There is some kind seed or flower thing sticking out of it.
This is a Tillandsia, one of the grey specimens (there are green ones as well in the genus).
In the US, Spanish moss, (Tillandsia usneoides) is one very famous variety that is native to Texas and other southern states. But yours is a variety of ball moss, either T. recurvata , (more photos), or T. baileyi.
Although tillandsias may look like moss or lichen, they are no parasites and won't harm the tree they grow on, they merely use it for support. This becomes very obvious when one sees them making no distinction between live trees and wooden telephone poles or even the wires.
If you want to keep your specimens, I'm actually a big fan of "leaving where they are", or in other words, to mimic the environement where you found and took them from as closely as possible. If you have a suitable tree in your garden, just hang the branches up in the tree so that they will have dappeled shade from the leaves in the summer and get watered by dew and rainfall and fertilized by the dust that blows on them.
If you insist to take them inside, note that they need enough light, but little direct sun, dilligent but careful spraying to mimic dew and once in a while a gentle "rain" in your shower. Don't bother with fertilizer.
Do not - I repeat, do not take the clumps of plants apart. You are very likely to hurt the plants and to re-mount them elsewhere is very difficult. Those plants don't want "repotting" at all. Some sources on the web suggest using a glue gun to place tillandsias on your chosen carrier. But they usually don't tell you that a standard glue gun will burn your plant just like it burns your finger. There are "low-temp" guns for plants on the market, but I can't give you qualified information on them.
I also would hesitate to place a greyish tillandsia in a terrarium, especially if it's mostly closed. The rule of thumb for grey ones is that they need a somewhat drier environement than green ones: the silverish fuzz is like a natural protection from the sun.
Note that tillandsias are somewhat "slow-motion" plants. They won't look much different from normal even when they are suffering already. My favourite counter-example is the peace lily: forget to water it, and it folds down all the leaves, but once you water it, it will recover within a few hours. Tillandsias don't droop, but at some point, they just shrivel up and die. They also don't give obvious signs of overwatering, but once you see moldy spots, it's too late.
I recommend that you use your environement to teach you: Take lots of walks and observe: how much light do the wild tillandsias get in those trees? Over the seasons? How do they look? How do they change during hot and dry spells or during times of frequent rain? Then compare your observations with the appearance of your plants at home, adjust your care regimen, if necessary.