There are three big, popular price checking and trade addons. This post aims to help you understand the differences between them and their respective advantages and disadvantages.
ATT scans the sales history of all guilds you belong to and uses that information to produce its price points. It is one of two addons that use data in this way, and as a result display only data from real purchases. These addons use a smaller sample size, but their pricing can be more accurate. These addons work best if you're in at least one large trading guild.
Advantages of ATT:
ATT prices might be inaccurate if someone in a guild you're in listed an item for far above or below market value and successfully sold it.
The philosophy behind MM is essentially the same as ATT. It scans the sales history of the guilds you're in and builds its pricing data from that. MM is much more mature than ATT, but it is also much slower and does introduce some login lag. It works best if you're in at least one large trading guild.
Advantages of MM:
MM prices might be inaccurate if someone in a guild you're in listed an item for far above or below market value and successfully sold it.
A new and arguably better alternative to TTC. Collects data in the ways that ATT and MM do in addition to the way that TTC does. A great all in one if you only want one addon that just shows you the price.
For your convenience, here is a link to their getting started guide.
TTC is both an addon with a client application, and a website. Players who use the TTC addon and run the Client.exe automatically scan every guild store they visit, and every listing they create, and the client synchronizes that information to the TTC website. The addon is then able to display current listing prices directly in your items' tooltips.
To run the client.exe, navigate to C:\Users\yourname\Documents\Elder Scrolls Online\live\AddOns\TamrielTradeCentre\Client and double click on Client.
You can visit the TTC Website to search listings all across Tamriel, but be warned that the item you're looking for may be gone by the time you make it to the trader.
TTC prices might be inaccurate if you fail to run the Client.exe, if the TTC website has received a lot of bogus item listings, or if players are artificially inflating or deflating item prices by listing far above or below market value on guild traders.
Advantages of TTC:
ATT and MM use the sales history of the guilds you're in to create their price data. Enter LibHistoire, an addon on ESOUI / Minion. Make sure to follow its instructions on ESOUI "You can speed up this process" to pull as much guild history as possible, as often as possible, so that your MM or ATT have the freshest, most accurate data.