Packet sniffers allow you to see everything that is travelling on the network. Wireshark is the most popular and free and is easy to download and install.
With a packet sniffer, you can read in plaintext anything that is not encrypted.
These require that you have access to the network where the communication is coming from (either server or host). You cannot read packets on a network that you do not have access to. That is why you have heard that they read what is coming from your own computer.
But, it is possible to gain access to other networks. Wifi (public or hacked private) is a popular and easy means of gaining access to other's network traffic. Some attackers go to great lengths to physically install sniffers in corporate networks so that they can access this type of data.
HTTPS encrypts information between a browser and the server so that sniffers cannot read the actual content of the communication. For a site that has a mix of HTTP and HTTPS content, parts of the communication will be readable, and some will not.
This is easy to test. Download and install Wireshark and start a capture. Then launch a web browser and navigate to a website that is not HTTPS. Stop the Wireshark capture, then look through the data. You will find the actual text of the website in the capture.
Try again on an HTTPS website (google, for instance) and you will see that you cannot read the text from the website.
Now try the whole process on a mixed website, and you will see what is encrypted and what is not. It is possible that ads and other 3rd party content has not been encrypted, for instance.
Packet sniffers are the technical tools, but pre-packaged hacking tools use packet sniffers built into themselves to read this information and format it for malicious uses. Firesheep is the currently famous tool that can hijack someone's connection to Facebook and Twitter.