What has a `1s^1` orbital?
`1s^1` is a term that describes the location and number of electrons in an atom. Unlike the protons and neutrons in a nucleus, which are all concentrated and relatively immobile in a specific place, the electrons in an atom have a much wider degree of motion and a far larger region in which to be located.
Because of a variety of quantum mechanical properties, which I will not elaborate upon here, the electron in an atom cannot be said to "orbit" the nucleus in the same way that a planet orbits the sun, despite the fact that this is how electrons are commonly portrayed in layman depictions of the atom. While we cannot say that an electron is in a specific spot at any given time, we can say that it has an extremely high probability of being located in a particular region, which is, for our purposes, a very similar quality. Like planets, these regions ("shells") are described as circles that get progressively larger the farther they are from the nucleus. The most limited of these shells are the S shells, which can only hold 2 electrons. The "1" prefix describes the energy levels closest to the nucleus; thus, the 1s shell is the very first location outside the nucleus in which an electron may be located. The final "1" says that a single electron is located in this shell.
Since the shells must, as a generalized rule, be full before electrons can be added to higher shells, this means that there is only one electron in the atom being described. The default answer would be that this atom is Hydrogen. However, it is entirely possible, although not very common, for ions to possess this configuration as well; for example, a Helium+1 or Lithium+2 ion would have this configuration, although they would probably only exist in a laboratory or the interior of a star.
The orbital is actually 1s. 1s1 indicates one electron in the 1s orbital. For hydrogen, this is its only electron and would be the complete electron configuration. However, all atoms in the ground state will have a "1s1" electron but will also have a 1s2, 2s1, etc up to the total number of electrons.