What are some signs that the tooth decay is still active?

Sensitivity. Active tooth decay can have symptoms such as sensitivity to sweets, cold and hot , biting , brushing and flossing. By having yearly dental exams, your dentist can make sure you are treated when decay is early and small, and make sure you are free of decay.
Unless treated. Tooth decay unless it was treated by your dentist always harbors bacteria and is therefore "active", and can cause further damage.
Several... Signs of active tooth decay are: discoloration of the affected tooth, sensitivity to hot or cold and/or sweets. Some teeth don't display symptoms, that is why the dentist takes radiographs to determine carious activity in areas that are hidden and non-symptomatic. See your dentist regularly, get your teeth cleaned and don't worry.
Pain. Pain upon drinking cold water or eating something sweetened with sugar will usually cause pain in a decayed tooth. However, having exposed tooth roots due to gum recession can produce the same symptoms, even though there is no decay there. A dental exam is needed to differentiate between the two.