The Vela pulsar is a nearby young pulsar actively radiating radio to γ-rays. We present evidence in this work that the Vela pulsar is also interacting with its surroundings via a relativistic wind, which manifests itself as two different nebular structures. ROSAT PSPC observations of the Vela pulsar show that a 45 arcmin long collimated X-ray feature projects from the pulsar. We favor the interpretation that the feature is a 'cocoon' of heated gas formed when a jet outflow from the Vela pulsar interacts with the interior medium of the supernova remnant. This interpretation is consistent with the observed center-filled morphology and spectrum of the cocoon structure. Combined ROSAT + ASCA observations of the 'head' of the cocoon, the point where the jet is believed to interact with the supernova remnant, demonstrate that the spectrum has a thermal peak near 1 keV, but extends to at least 7 keV. No distinct spectral lines are seen. The spectral parameters of the cocoon could be produced by a cocoon with a pulsar jet whose speed is at least 800 km s-1, depending on the angle of inclination. The mechanical power driving the jet is ≥1036 erg s-1, consistent with the Vela pulsar's rotational energy loss rate. On smaller spatial scales, it has been known that the Vela pulsar is surrounded by a 2 arcmin diameter 'compact' nebula which has power law spectral emission. Our ROSAT HRI observations of the region show that the nebula very likely a bow shock structure formed by a nearly isotropic pulsar wind interacting with the supernova remnant. The axis of the nebula is aligned with the pulsar's known proper motion vector. The high particle energies and magnetic fields near the pulsar make the bow shock an ideal environment for generating X-ray synchrotron emission. We show that a full three dimensional model of the nebula, taking into account what is known about the geometry and pulsar wind physics, is consistent with the observations.