I would like to invite you 'for' TO (you invite people To the cinema, to the theatre itp) my 18-'birthday's BIRTHDAY party.
'On' AT this party wTHERE ill be competitions, amusements 'for which' (nie, nie tak - napisz WITH) great awards.
'Presence pleasantly seen' (to jest bardzo dziwne tutaj ..moze I will be pleased if you can make it...
'Peter invites' (nie rozumiem tego na koncu zaproszenia - lepiej niepotr)