Studdy behavior in horses means acting like an intact male. It's most often used when you have a horse that is not, in fact, an intact male, but is behaving like one - flirting with females (and sometimes other males), being territorial and aggressive or prancing around in a particular way stallions do.
Geldings that were cut after puberty may have episodes of "studdy" behavior. If a mare acts "studdy" then it often means she has a hormonal imbalance (It's normal for a mare to show flirtatious behavior when in heat, but studdy behavior includes "snaking" the neck, trying to round up the other horse, tec). Some mares may also act studdy when around 90 days pregnant, regardless of the sex of the foal - sometimes to the point of having to be separated from other horses.