I'm Indian, so I can answer this:
"Chai" is simply the word for "tea" in India, and "masala" means spices. So masala tea or masala chai is tea that is made usually with milk, mint, cardamom and sometimes a couple of other spices.
Having said that, in western cultures, like here in Canada, people don't usually know that "chai" means "tea", so when somebody says chai they are usually talking about masala tea. This is the reason why the term "chai tea" is very common, even though it translates to "tea tea".
In summary:
For Indians, chai is any kind of tea, and masala tea is masala chai, which is spicy tea.
In western countries, chai, masala tea and chai tea all refer to the same thing. The types and amounts of spices can vary depending on who's making the tea.