The cups on a full cup bra completely cover the breasts. They offer the most support of any type of bra so are a popular choice for women with large cup sizes. As they are designed for women who want more support and comfort, the shoulder straps are usually wider too. Sports bras should always have full cups.
If you have trouble with bulging over the top of lower cut bras then try full cups. The cup comes up high enough that they will fully contain your breasts so you will get a smooth line.
The main downside to full cup bras is that they will be visibile with anything that comes down even slightly low, all this means is you need to make sure you own a couple of other bra shapes like a demi or plunge to suit different clothes. Anything with a higher neck, like most t-shirts or sweaters will be fine with a full cup bra.