Yes, you can work on your own plane, if:
I. You are performing maintenance work under proper supervision, or
II. You are performing preventive maintenance, and each of the following are true:
- You are a certificated pilot (except sport pilot),
- You either own or operate the aircraft in question, and
- The aircraft is operated under part 91.
14 CFR 43 regulates aircraft maintenance for those aircraft with a U.S. airworthiness certificate, but not for experimental aircraft (see §43.1 Applicability). For the purposes of this answer, I will use the term "aircraft maintenance work" to refer to any and all maintenance work on U.S. certificated aircraft that might fall under the legal definition of maintenance, preventive maintenance, and alterations. This does not include inspections.
§43.3 regulates who is authorized to perform aircraft maintenance work. Aircraft maintenance work is subdivided into maintenance, preventive maintenance, rebuilding, and alterations. For the purposes of this question, we are interested in two of the many description of who may perform aircraft maintenance work.
The first is found in §43.3 (d), emphasis mine:
(d) A person working under the supervision of a holder of a mechanic
or repairman certificate may perform the maintenance, preventive
maintenance, and alterations that his supervisor is authorized to
perform, if the supervisor personally observes the work being done to
the extent necessary to ensure that it is being done properly and if
the supervisor is readily available, in person, for consultation.
However, this paragraph does not authorize the performance of any
inspection required by Part 91 or Part 125 of this chapter or any
inspection performed after a major repair or alteration.
In other words, anyone can perform aircraft maintenance work under proper supervision. In your case, if you are working under an authorized mechanic who properly supervises you and signs off the work, yes, you can perform aircraft maintenance work.
The second is found in §43.3 (g), emphasis mine:
(g) Except for holders of a sport pilot certificate, the holder of a
pilot certificate issued under part 61 may perform preventive
maintenance on any aircraft owned or operated by that pilot which is
not used under part 121, 129, or 135 of this chapter. The holder of a
sport pilot certificate may perform preventive maintenance on an
aircraft owned or operated by that pilot and issued a special
airworthiness certificate in the light-sport category.
This is more complex. If we are to assume that you are the holder of a pilot certificate issued under part 61—for example a Private Pilot certificate—then you meet that part of the qualification. If you own your own aircraft, in whole or in shares, then you also meet that part of the qualification.
Now, you may also qualify if you only operate the aircraft regardless of ownership. 14 CFR 1.1 defines Operate as follows:
Operate, with respect to aircraft, means use, cause to use or
authorize to use aircraft, for the purpose (except as provided in
§91.13 of this chapter) of air navigation including the piloting of
aircraft, with or without the right of legal control (as owner,
lessee, or otherwise).
(Note that §91.13 defines careless or reckless operation. In other words, if you use an aircraft carelessly or recklessly you are not actually operating it according to the legal definition and so do not have preventative maintenance privileges for that aircraft. Now you know.)
So, if you own, pilot, or otherwise operate the aircraft, you may have preventive maintenance privileges. The final qualification will be whether you operate the aircraft under part 91—which would give you preventive maintenance privileges—or under parts 121, 129, or 135—which would generally deny you preventive maintenance privileges. Note that parts 121, 129, and 135 refer to scheduled or on-demand passenger or freight operations available to the general public, to put it simply.
Now, some of the answers here imply that you are not authorized to perform preventive maintenance if the aircraft you own or operate is used for commercial purposes.
This is not the case.
Many commercial purposes are allowed under part 91, including rental, flight instruction, aerial patrols, sightseeing, business transportation (think business jets), skydiving, and glider or banner towing, among others. If you otherwise meet the above qualifications and operate an aircraft for such purposes, you are legally authorized to perform preventive maintenance.
This leaves us with the question of what constitutes preventive maintenance. The legal list is found in 14 CFR 43 Appendix A (c) and other answers have done a good job of outlining that, so I won't go into it further.