This is perhaps one of the most misunderstood and abused of all common English words. Properly, a thing purports that which appears to be, or is conveyed as, its essence, import, or effect, as, for example, an act that purports benevolent purpose, or a document that purports authenticity. It may also serve as a noun, the purport of a thing being that which it purports, but this usage has become less common outside of legal jargon.

The word does not imply any commitment to the verity of the attributed quality; indeed, it is often intended to cast doubt:

Perinatal mortality rates do not contain what they purport to contain. *
The subject may be a person rather than an inanimate object, e.g., one who purports purity of motives. However, in most such instances, unless an essential characterization is intended, a better choice is available.
The president is, of course, on stronger ground when he purports to be acting pursuant to an act of Congress. *
Clearly, the action in question here is not central to the president's being, and therefore is not his purport. Depending on the intended sense of the verb, a more judicious choice might be appears, professes, or pretends.

As illustrated in both citations above, it is common for a purported attribute to be formulated as an infinitive phrase. This is a legitimate and useful device for generating a substantive, but in this context it tends to be overworked in contemporary usage:

We are aware of a teaching system that purports to be of research quality and purports to be fully portable.*
And every idea in the universe which purports to be of a religious or philosophical significance must fall under one or other of these two religious categories. *
In both examples above, the infinitive is superfluous: in the first, the teaching system may be described more directly as one that purports research quality and full portability; in the second, the idea in question simply purports religious or philosophical significance.

In any case, the direct object—that which is purported—is the attributed quality and not the thing to which, or the person to whom, it is being attributed. In particular, purport is not a reflexive verb:

The first of the three presidential debates in Boston last night showed the world that Al Gore is not always the great debater that he purports himself to be. *
Occasionally, it appears as a malapropism for comport:
In the sporting world, it's usually how a player purports himself on and off the field and how he deals with adversity and pressure situations that make him worthy of adulation. *

A more common blunder is the mistaking of purport for a fancy variant of report:

Why does Philip Morris continue to purport that ventilation systems will clear the air of tobacco smoke? *

Similarly, a news story under the following headline reveals that it is the examination being studied, rather than the study itself, that appears to be biased:

Study Purports Bias Against Minorities in LSAT *

This confusion often leads to the misuse of purport in the passive voice, which muddles both subject and object:

Developer of Purported Breast Enhancement Product Settles FTC Charges *
No, it is the power to enhance breasts that is purported, and the product that is doing the purporting.

It is difficult to imagine a proper use of the adverbial form, which is commonly taken as a synonym of reportedly:

The Saudi branch of the Al-Qaeda extremist network has purportedly claimed the killing of an Irish national in Riyadh earlier this month, in a statement published on a website on Friday. *
Chechen rebel leader Shamil Basayev purportedly took responsibility Friday for a bloody school siege and other recent terrorist attacks that have killed more than 430 people, but put ultimate blame on Russian President Vladimir Putin. *
The future of this battered word appears hopeless—now bleaker than ever as we read the following in a column by William Safire, a writer known for his careful use of language:
But it turns out CBS had only read Maj. Gen. Bobby Hodges the purported memos on the phone, and did not trouble to show them to him. *