I am interested to know when would one use these lenses and what are they best suited for?
The Petzval lens design was revolutionary, because it was the first commercial lens with a fast aperture. Previously, the Chevalier lenses had a fixed aperture of f/15, and when combined with recording media with a sensitivity equivalent to a small fraction of today's ISO 1, that meant very long exposures.
Chevalier improved on this with an f/5.6 design, but that suffered from poor sharpness. Petzval's design allowed apertures up to f/3.6, and had great sharpness in the center, making it a much better tool for portraits.
So, that's interesting historically. If this aspect interests you, there's a fantastic and detailed article at antiquecameras.net, with vintage advertisements and pictures of several vintages lenses, although unfortunately not many samples. You can find samples in this Large Format Photography forum thread.
Additionally, the design has very dramatic field curvature, which tends to render the background as a dramatic and distinctive swirl — the "Petzval swirl". Like all such optical effects, this is hard to replicate exactly with a filter without a 3D model of the scene, and if you like the look and are fascinated with the low-tech, lo-fi aesthetic that the Lomography company (and others like Lensbaby) have made their bread and butter, this may be something you'd like to add to your creative options.
Petzval lens, developed by Joseph Petzval in 1840, in Vienna, is a lens consisted of two doublet lens.
It was popular in the 19th century since most of the photos were shot using theses lenses. Photos are immediately recognizable for their sharpness and crispness, strong saturation, swirly bokeh effect, artful vignettes and narrow depth of field. They are extremely ideal for portrait photography which make it really really famous during those times.
Modern lenses are designed to minimized field curvature which will have the whole image in focus. In Petzval lens, it's different. It has an optical effect that gives you an area where it is sharply focused over a narrow field. The there is a vignetting effect towards the non-focused area.
Click here to tell know more about the Petzvel Lenses or View sample photos using Petzvel.