I am aware that circular DNA can be both relaxed and super coiled. However when running an agarose gel of the circular plasmid along with singly digested plasmid with BamHI and HindIII, I see 1 band for the linearized plasmid (lanes 2-5) (as expected) but 5 bands for the circular plasmid (lanes 6 and 7).
I don't have hands on it, but I will not be surprised if supercoiled DNA migrates at different distances according to some inner topological conformation (i.e., more or less coiled AND/OR more or less nicked). Similarly, this picture highlights >8 conformations. What is run in the gel is circularized phage DNA with some degree of knotting due to the circularization (phage DNA is otherwise linear).
Another theoretical possibility is that two or more plasmids of different sizes were grown in the same maxiprep because of some contamination. But in this case, you will spot the problem also in the digestion, so it does not seem to be your case.
Gianpaolo is correct that you are seeing the different conformations of circular DNA.
Growing plasmids up in bacteria produces 3 main forms: relaxed, coiled and super-coiled.
Relaxed tends to run a little higher than the expected size because it cannot travel as efficiently through the agarase;
Coiled migrates a little farther than the expected size because it is a little more compact and therefore migrates more easil;
Super-coiled, the most compact, migrates much farther than the expected size.
I would conclude from your picture the following:
That the HindIII cuts are efficient, and have linearized nearly all of your plasmid.
BamHI is cutting, but not as efficiently as HindIII for (presumably) the same amount of time. This should be digested longer if it's really important to digest with just BamHI, but for diagnostic purposes, it is fine. I say this because of the faint banding seen at ~4.5 and ~3.7 kb, which are likely your coiled and super-coiled plasmid DNA, respectively.
From your linear digests, the plasmid is 6 kb.
Looking at undigested control plasmid, your method of mini prep is not clean and looks to be contaminated, possibly by genomic DNA. Also, you see hints of streaking in the single digest lanes. Was this done using trizol?
I would expect the band at 10 kb to possibly be the relaxed form of the plasmid, with much of the plasmid DNA existing in a super-coiled conformation.
If these are important samples, I would consider purifying them using cleaner methods, for example using a column.
I think these answers have all missed the most likely explanation of the multiple bands in the uncut sample. I agree that you are likely to see supercoiled, linear and relaxed (open) circles in that order going up the gel. But then all of those higher forms are the same thing being repeated for dimers, trimers etc. formed by recombination between plasmid molecules. So, for example, I think the 10 kb band in the uncut lanes is a supercoiled dimer. I have a dim memory that you see more of this sort of thing if the DNA is prepared from a recA strain of E. coli.