The use of this suffix has grown recently. It can irritate speakers of traditional British English that a distinction is being lost. (See English Word-Formation p. 243, and Burchfield's Fowler) If you are writing academic English in the UK, take some notice of what follows: if your teachers appear to mind, observe their prejudices and do as AWE says. (Even if they do not appear to mind, you may like to become more precise and traditional in your usages anyway.)
-ee represents the passive participle in French ([verb-stem +]-é for the masculine; [-ée for the feminine]), and names a person (or thing): 'a person who is/has been [verb]-ed'. Often, in English, it is a passive formed from the indirect object - 'a person to [or for] whom something has been [e.g.] given'. In banking, for example, a payee is someone to whom the cheque will be paid - the person who will get the money. The person who has to pay the money should be a payer (for more about the difference between these two suffixes, see -ee - -er).
The development in usage which irritates traditionalists is the use of -ee where -er (meaning the agent, 'the person who --s') would once have been expected. A convict who breaks out of jail, logically, should be an escaper, and often is; why then should it be necessary, asks the traditionalist, to name such a criminal an escapee? But this word is recorded in OED from as early as 1875-6. (It may be a translation from French, where the rules governing the use of the participle are different: in French, the literal translation of "he has escaped" is "he is escaped", and of "an escaped prisoner" (or "someone who has escaped") is simply "an escaped".) This particular use of the suffix -ee is more usual in American English. The word 'indictee', used on a news broadcast recently to mean the person facing an indictment in court, is an unnecessary replacement for the English word 'accused'. Attendee, for someone who is attending a meeting, seems to be of US origin; it belongs to the second half of the twentieth century. As long ago as 1880, Webster defined standee as 'one who is obliged to stand': UK traditionalists like to think that a standee should logically be 'a person on whom someone is standing', and there has been comment on the subject in the Guardian newspaper in 2005 (see Smallweed 16-04-05). The idea could be stretched to say that an attendee should be someone on whom an attendant is waiting -- as someone eating at a restaurant is the attendee whom the waiter is attending.
Only use words ending in '-ee' if you are sure they are needed - or if your teacher has already used them.
See also -é - -ée for a note on words ending in -ee that still have an accent.
(There is a long tradition of humour around the suffix. In 1754, the novelist Samuel Richardson wrote of a 'lover' and a 'lovee'; and another novelist, Laurence Sterne, used 'jestee' for a person on whom a joke had been played. In general, in academic English, you should avoid attempts at humour.)