Can anyone explain the constructions ge + verb + en (e.g. gesehen, geschlafen ) and ge + verb + t (e.g gespielt)?
A brief and incomplete summary:
The forms you mentioned are past participles. As in English, they are used for the present perfect and the past perfect (e.g. I have SEEN, he had PLAYED). As in English, there are regular and irregular past participles.
In English, past participles of regular verbs are formed by adding the ending "-ed" to the infinitive (I have playED). In German, the past participles of regular verbs are formed by using "ge- + infinitive stem + (e)t". E.g. : machen (to make) = infinitive. Infinitive stem, i.e. infinitive minus -en = mach. Past participle = ge-mach-t (= made).
As in English, some verbs and their past participles are irregular (cf. English "I have SEEN"). In German, past participles of irregular ("strong") verbs are formed by using "ge + past participle stem + en". The past participle stem can be identical with the infinitive stem as in your examples (gesehen, geschlafen), but very often the vowel changes, e.g. infinitive: "singen" (to sing) vs. past participle: "gesungen" (sung). For this reason, the past participles of irregular verbs have to be memorized.
Adding to what Katherle wrote, verbs of Latin origin do not take the prefix ge-. The past participle of these words is formed by adding -t to the stem: studieren -> studiert; lamentieren -> lamentiert; addieren -> addiert; subtrahieren -> subtrahiert; immatrikulieren -> immatrikuliert