There are many different rhythm names which teachers use, which are usually based on the French time names (ta for crotchet, ta-te for quavers and so on).
Kodaly adapted these and Kodaly teachers tend to use ta and ti-ti or te-te. Some people use tiri-tiri for semiquavers but some prefer tika-tika (I'm one of these!).
They should always be spoken to a steady pulse. They really work, both for rhythm reading and for identification of the rhythm you hear.
Flash cards are just cards with various rhythm patterns written on (sometimes just the 'sticks' are written without the note-heads). They are read by the student using the rhythm names. If the teacher moves to the next card on the last beat of the previous one you are encouraging the student to read ahead, fluently. This can then be developed into changing the card one whole bar ahead so that the student is reading in canon! Body percussion can also be used (eg click the crotchets, tap the quavers, touch your nose in the rests etc).
As Nat and Helen discovered in David Vinden's class last week, they can also be used to help memorisation! He asked the class to read 4 bars of 4/4 rhythm with rhythm names, then turned over one card at a time until the whole 4 bars were memorised.
There are many other variations on these activities and games! Children can choose a card and improvise a vocal or played melody to it, for example.
They really do work and enable the student to just concentrate on the rhythms without also having to worry about the technique on the instrument at the same time.
I hope this explains it a bit!