I would like to know what species this plant is, I think either Crassula ovata or Portulacaria afra:
Also: What are the little white thingies coming out of the branches?
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That's a Crassula ovata: the leaves are more elongated (ovata = "egg-shaped") and larger than Portulacaria afra. Also, they are missing the indentation at the tip that makes Portulacaria afra leaves heart-shaped. The growth of Crassula is "stockier" and the branches are smooth. Find a picture of a Portulacaria in this question for comparison.
The white "fibers" are aerial roots: if parts of the branches touch the ground or if you break off a branch and place it on soil, it shold root quite quickly.
Crassula ovata yes, but, whilst aerial or adventitious roots on this plant are more commonly produced in humid atmospheres, they do not usually produce so many, and this is likely the plant's attempt to 'escape' and root itself elsewhere, away from the pot. I suspect it's doing so because it's growing in difficult conditions - maybe it's rootbound, needs a new, larger pot, which from what I can see in the photograph, may well be the case. I suggest you turn it out of its pot to see what condition the roots are in, and if they are overcrowded, pot up into something larger.