The Prince of Wales here is the notorious Prince Regent (later George IV), and the Princess of Wales is his wife, Caroline of Brunswick-Wolfenbüttel; they had become estranged within a few years after their marriage in 1795. She won much public support due to his treatment of her (and from widespread public distaste for the Prince Regent generally), even though she committed some indiscretions and infidelities herself. However, he was able to exclude her from official privileges (though not to fully divorce her, which would have required the assent of the hostile parliament), and his accession to the regency, and later to the throne, was not affected. (See also sex in Pride and Prejudice and the double standard.)
[The following written 1995:] Of course, this quaint historical sidelight has absolutely no contemporary relevance whatsoever. No way!
"I suppose all the World is sitting in Judgement upon the Princess of Wales's Letter. Poor woman, I shall support her as long as I can, because she is a Woman, & because I hate her Husband -- but I can hardly forgive her for calling herself ``attached & affectionate'' to a Man whom she must detest -- & the intimacy said to subsist between her & Lady Oxford is bad -- I do not know what to do about it; but if I must give up the Princess, I am resolved at least always to think that she would have been respectable, if the Prince had behaved only tolerably by her at first. --"
A contemporary caricature of the Prince Regent:
[P.S. -- Yes, I'm aware that the coat of arms in the center of the above image (if you have a graphical browser) was actually not used on its own until 1837.]