The Global System for Mobile Communications, or GSM network for short, includes many popular phone carriers, such as AT&T and T-Mobile. The only phones compatible with T-Mobile are those compatible with other GSM network phones, because all information on a T-Mobile phone, such as your contacts, text messages and apps, are stored on a SIM card. The Sim cards on all GSM phones are compatible with each other, so you can use a T-Mobile phone on an AT&T network, or any other GSM network, by swapping out the respective SIM cards through a process called unlocking.
The Code Division Multiple Access, or CDMA, network is another widely used network for mobile phone service providers in the United States. Sprint/Nextel and Verizon are on this network. The phones designed to be used on CDMA networks don't include SIM cards, or SIM card placeholders. Instead, these phones store contact information, text messages and apps on the phone itself. Phones distributed to CDMA network providers are locked to the carriers that they're shipped to; you must use the service provider who sold you the phone.
Unlocking is the process that allows you to remove the SIM card from a phone on one GSM provider, and use it on another. You don't even need the cooperation of both providers to carry out this process. Often, your current GSM network provider will give you an unlock code for free, or for a small fee, depending on the company's policies. By unlocking your new phone, you can insert your old phone's SIM card into it, and it will instantly contain all of your contacts and information. The only difference is that you'll have a new service provider.
Because Verizon uses the CDMA network, and there are no SIM cards used on the CDMA network, you cannot simply insert a T-Mobile SIM card into an LG phone — or any Verizon phone — and use the new service provider. If the LG phone was on AT&T, or a different GSM network, this wouldn't be a problem. But because these two providers are on two completely different networks, the procedure won't work. However, it is possible to port your T-Mobile phone's information to your new Verizon phone, but this must be done by an authorized Verizon dealer. You'll still be able to retain your phone's information, but it's a process that takes time, and may even cost money, whereas unlocking takes only minutes, and can possibly be done for free.