Savings fares, eg Dauer-Spezial (unfortunately only in German) or Sparpreis fares, require an express train (ICE/IC/EC) connection, although there can be regional connection(s) at the beginning and destination. There are no starting time limits on Savings fares, just that they must be offered on that connection. Savings fares must be purchased in advance, usually online.
Länder-Tickets, which are offered by the state (as opposed to Federal) run, local train organizations (Land = State, Länder is plural of Land), are only valid on regional trains (IRE/RE/RB/S and some other private companies, like ALX) in that state. Länder-Tickets have a 9 AM weekday limit. Länder-Tickets can be purchased at an automat or ticket counter (€2 extra at the counter), have no advance purchase requirements (so can be purchased the day of travel), and are unlimited (so they don't sell out). Yes, travel time is often longer.
Standard tickets are open tickets, usable on any trains within a time limit (eg 30 days). If you buy reservations (~€4) with standard tickets (only on express trains), they are only valid for that train and expire if you don't use them, but the ticket, itself, is still valid.