Given elroy's examples I will limit my answer to active participles.
Let's start with compound verbal forms. Besides of the simple forms in elroy's examples there are also complex past forms which recursively employ verbs být (= to be) and bývat (= to use to be) as auxiliaries:
byl bych lovil
býval bych lovil
byl bych býval lovil
byl bych býval byl lovil
byl jsem lovil
býval jsem lovil
The conditional forms are still in common use but even native speakers mix them up.
The unconditional complex forms are archaic. Average native speaker is unfamiliar with these forms.
Now, Czech is a pro-drop language which means that all these complex forms may be shortened by dropping the parts which are inferable grammatically or from the context. In modern Czech auxiliary is always dropped for singular third person, hence the form lovil (= he hunted) instead of the original old Czech jest lovil.
The other cases of dropping parts of the compound verbal form are optional, common especially in spoken Czech (strict teachers may ask you to speak in complete sentences). For the basic compound forms it is always the auxiliary which is dropped, i.e. you end up with the participle alone:
Viděl jsi ho? Viděl! = Did you see him? I did!
Šel jsi nebo (jsi) běžel? Běžel! = Did you walk or (did you) run? I did run.
For the more complex compound forms there may be more ways to drop some parts. You may end up only with the participle of the verb of meaning, but also with the participle of the auxiliary:
Býval bys lovil? Býval!
Býval bys lovil? Býval bych!
Býval bys lovil nebo (bys býval) šel domů? Lovil!
As for the usage which is not compound verbal form, it kind of depends on how you define participle in Czech.
The participle in compound verbal forms is originally an adjective expressing verbal action without respect to verbal tense. As part of the compound verbal form it started to express past action.
To make things more complicated, there are two sets of adjectival forms in Czech - short form (original one) and long form (newer form which is a merge of the short form and definite articles). Participles in compound verbal forms are of the short form, while almost all modern adjectives are of the long form.
There are also long adjectival forms derived from the short participles and it is matter of convention how you classify them. You may see them as fresh new adjectives, you may see them as long forms of the participle, you may classify them either way based on some additional criterion (e.g. existence of comparative form or existence of secondary meaning).
And finally, there are nouns derived from all these adjectival forms. Either nouns whose form is simply identical to the adjectival form, or nouns formed using some suffixes.
It's up to you where you draw the line between participle and other related forms.
Some concrete examples:
Adjective jedlý (=eatable/edible) does not reflect past action, i.e. it is the long form of the original adjectival form based on verb jíst (=to eat).
Adjective (participle?) opilý (=drunk) describes outcome of past action, i.e. it is the long adjectival form corresponding to participle opil.
Some widely used adjectives related to participles:
minout (= to pass) => minulý (= past)
bývat (= to use to be) => bývalý (= former)
zrát (= to ripen/mature) => zralý (= ripe/mature)
dospět (= to reach/mature) => dospělý (= adult/mature)
dokonat (= to end/finish/complete/perfect) => dokonalý (= perfect/complete)
Some of these adjectives may be used as nouns:
dospělý = an adult
bývalý = ex-partner/ex-husband
pan Dokonalý = Mr. Perfect
And finally, there are the very short forms of the participles (or short forms of the original adjective?) turned into nouns:
1) l-participle -> masculine noun
characters:
břídil = bungler
loudil = someone who is scrounging
loudal = straggler, slowpoke
škudlil = someone who is skimping / saving too much
common surnames (this is actually very common usage!):
Hasil, Kroutil, Nešetřil, Vyskočil, Pospíšil, Nedbal, Zbořil, Stejskal, Kvapil, Sekal...
hejkal = forest deamon known for hooting
mýval = racoon
rozrazil = veronica (plant)
2) la-participle -> masculine noun
surnames: Váhala, Prchala, Zdráhala
Sometimes, a whole phrase with participle can turn into a single word:
draho koupil -> Drahokoupil (surname) = somebody who bought something at high price
po tmě šil -> potměšilý (adjective) or Potměšil (surname) = crafty, cunning, malicious (literaly: the one who is sewing at nigth / in the darkness)