Hello, spaceshuttle!
Your question is very legitimate in my view. Pooja is worship of the simpler form not using the sacrificial fire. Pooja is of many kinds, from the condensed one to the most elaborate rites. Usually the pooja is considered as ShODaSOpacAra pooja (or, worship with 16 - ShODaSA - services or upacAras).
I will give in a nut shell these 16 'services' and very short comments on each:
01. Asanam - offering a seat
02. pAdyam - washing the feet with water (the word pAdyam means water for washing the feet)
03. arghyam - this word means 'something valuable' but over the millennia, it now stands reduced to a simple spoonful of water; in the vedic past a respectful oblation of cooked rice, dUrva grass, flowers and water was given into the sacrificial fire, I understand.
04. Acamaneeyam - water for gargling (cleaning the mouth)
05. madhuparkkam -this used to be a mixture of honey (madhu in sanskrit) and curd (arkka) and sometimes also ghee (clarified butter)
06. snAnam (abhiShEkam) - here the deity/deities who are worshipped are given bath ceremonially along with chanting of appropriate mantras.
07. snAnAnantara Acamaneeyam - water for gargling, after bath.
08. vastrOttareeyam - a pair of cloth, one to be worn on the waist and the other to cover the torso or upper half.
09. yajnOpaveetAbharaNam - the sacred thread and jewelry (for both male and female deities, but in the case of female deities, nowadays the priests omit yajnOpaveeta)
10. gandhAkShatA - unbroken polished rice which has been coloured with turmeric powder and perfumes like sandal paste, kasturi, gOrOcana, etc.
11. puShpArcana - doing arcana with flowers
12. Dhoopam - the perfumed smoke produced by sAmbrANi or benzoin resin; there are other herbal mixtures also.
13. AlankAradeepam - decorated and lighted lamp being waved around the idol ceremonially; originally this must have been the giving of a light to the honoured guest in case he/she was to return after nightfall; in Kerala even today a guest who stays for the night meals will be given one or two bunches of dried coconut leaves, free, which will serve to light his way back.
14. naivEdyam -food to eat
15. tAmboolam - pan, betel leaves, arecanut only are used; chuna or slaked lime is avoided.
16. karpoora neerAjanam (popularly known now as "Arati") - ceremonial waving of lighted camphor around the idol, from head to feet and back again.
I am only a novice. There are many stalwarts who may kindly correct, edit and modify the above. I will only be grateful to them.
Homams are more elaborate and there still is some reservation in the brahmin mind that only those with the sacred thread (brAhmaNa, kShatriya and vaiSya) ae eligible to perform homams. Essentially in a homam, the deva/s are called and welcomed and oblations are offered into the sacrificial fire with mantras ending with the chant "swAhA" (meaning welcome). There are very many kinds of homams and even a large tome may not be sufficient to describe all of them. regarding Ganapathi Homam I shall write tomorrow pl.