Carriers run off 2 different software technologies.
CDMA Technology: Verizon and Sprint both run off CDMA technology. However, Verizon devices will not work with Sprint services and vice versa.
GSM Technology: AT&T and T-mobile both run off GSM technology. Unlike CDMA, GSM carriers are compatible with one another. An AT&T phone will work on T-Mobile services. T-mobile services will work on AT&T
If you are wishing to switch carriers, you are more versatile with your device if it runs off GSM software technology.
Not to confuse things, but some newer model devices like iPhone 6, 6+, 6s, 6s+ and Google Nexus 5 models and Nexus 6 models are compatible with all carriers. Newer model phones have been wired to be compatible with both technologies.
Switching carriers - the carriers outside of the big four all run off one of the big four's networks. If you have a Sprint phone for example, you will only be able to switch to a carrier that runs off the Sprint network (in general).
T-Mobile and AT&T run off GSM technology - that means its easy to switch between the two and keep your phone - or you can switch to any other carrier that runs off the AT&T or T-Mobile network. In our search options you can select "Bring your own GSM phone" - that means we will show you all results from carriers that work on GSM and you can easily switch...
Here is a link to the Bring Your Own Phone BYO GSM carriers and plans
If you are wishing to switch carriers, check out this WhistleOut Guide on Switching carriers.
Here is another guide specifically about if Verizon/Sprint will work with each other.