You will need some form of photo editing software that will allow you to do the basic photo adjustments. It really needs to allow you to mask and layer images. This means something like Photoshop or Paint Shop Pro (now Corel). Many of the major camera manufacturers are supplying ever increasing feature rich editing software with their cameras. They incorporate things like panorama software. What I have seen of some of the software they lack the ability to take the editing process further by saylayer/masking images and therefore you may find that you end up using tried and tested routes if you want the full freedom of editing and assembling your images
For Mac users Apple's Aperture is becoming popular and Adobe's Lightroom. It really depends on whether you are taking just straight shots or wanting to take panos and the like.
Panoramic software is generally based on Pano Tools. This suite of mini programs converts and blends images to a format that will let the user spin and manipulate the image in a viewer. The viewer does not necessarily have to be used in a browser but is usually done so. The program is not really main street user friendly it has to be said. I would recommend one of the GUIs like Hugin, PTMac or PTGUI. These programs seamlessly combine the elements of Pano Tools and lead you through the process of producing a panorama.
The panorama does not have to be a complete spin in which case you will not need a veiwer. Sometimes however you can view partial panoramas in a viewer and they do look good. But for full 360 degree panos you are going to need a viewer. There are plenty of free viewers indeed the assemblers mentioned above have viewers. Like everything though there are unique ones and I use Pure Tools & Player Pro from Immervision mainly because it allows a high level of customization, it is intuitive to use, it allows a full screen experience and is very stable. Lastly it does not cost a fortune. Other features include Flash and Quicktime output. The output can also carry XML so it becomes highly interactive if you wish.