I recently started trying to understand the theory of relativity and I have the following question. Because earth is orbiting the sun at approx 67,000 mph (and its my understanding time slows down the faster you are moving) what are the effect of time on earth compared to a completely stationary observer watching earth's orbit, from outer space? How would the twin paradox work with one twin on earth and the other twin completely stationary in outer space?
Time dilation is a relative phenomenon, not a local phenomenon.
As adrianmcmenamin mentioned, Earth's motion around the sun is cause for no more time dilation than any other point in the universe. Consider that the sun is travelling around the galaxy at 200 km/s, and the Milky Way is approaching Andromeda at 130 km/s. We do not know at what speed we are approaching the great attractor, and I hope that at this point it becomes obvious that there is no difference in "us approaching the great attractor" to "the great attractor approaching us". Nothing is considered "stationary to the background" in an infinite, expanding universe as there is no "background" to measure against.
Consider the gedankenexperiment of comparing a clock on Earth to a clock orbiting the galaxy. Now compare those clocks to clocks outside the galaxy. You might consider the clock that ticks the most time away to be the "most at rest clock". Now I ask you, how will you compare the clocks? You must move at a significant portion of the speed of light for a very long time to get information from one clock to the next. Thus, the messenger would suffer the effects of Special Relativity, rendering the comparison meaningless.
To further the point, consider the development of International Atomic Time. I'll quote directly from wikipedia:
In the 1970s, it became clear that the clocks participating in TAI were ticking at different rates due to gravitational time dilation
Consider, those were clocks on Earth suffering from measurable time dilation due only to their latitude and altitude differences and the effect of moving information across the distances between them!
Relativity (special and general) is founded on the theory - which has been well tested - that there is no fundamental reference frame in space and hence there can be no "completely stationary" observers at all.
All we can say is that observers move relative to one another.
The time dilation effect of the Earth's motion at 67,000 mph relative to another observer is actually very small, being inversely proportional to the square root of 1 minus the relative velocity squared divided by the speed of light squared - essentially this is a ratio of about 1.000000005 - a second in every seven years or so.