Puja is the Hindu ritual of worshiping God in a systematic manner as prescribed in the Vedic scriptures. An idol or image or symbol of the deity is kept and worshiped by chanting the mantras & verses. The various names of the deity and the mantras are recited. The number of times they are recited depends on one’s convenience, capacity and time (Yatha Shakti). Some recite 108 times, or 1008 or more. Various offerings like fruits, flowers, camphor, sweets, milk etc. are offered. These are called Shodasa Upacharas. It is a way of expressing ones faith in God, a way of seeking salvation and a way of seeking His blessings to ward off the troubles and to ask Him to grant ones desired goals and objectives.
Homa is an ancient Hindu ritual performed to invoke Agni, the God of fire, by offering Ghee (clarified butter) and other sacrificial offerings. As Havih means Ghee and Havisha means offering and Yagyameans sacrifice, Homa is also called Havan and Yagya. After performing the Pooja, a fire is lit in a Homakunda as shown in the above picture. If one has done 1000 mantra recitals during the pooja, the 1/10 of that number, i.e., 100 times the mantra is recited again and with each mantra recital Ghee is pored into the Homakunda. Other offerings like cooked rice etc are also offered as prescribed. The ingredients vary according to the deity worshiped. Brahma is the creator of the world and the Sun God is His son. The Sun is the main source of energy and fire represents Sun. Hence ending the Pooja with offering to fire is considered very auspicious and enhances the beneficial effects of the pooja.
Technically a Homa and Yagya (also called Yajna) are the same but Yagya is a much larger scale Homa. To give you an idea: the famous Ekadasa Rudram where normally 11 pundits perform the Homa is called Rudra Ekadasa Homam. Where as the Ati Rudram, where 121 pundits perform the Homa is called Ati Rudra Yagya. So if you get a Havan/Homa done by a few pundits, you should not call it a Yagya. Today a lot of astrologers and websites are promising to perform “Yajnas” or “Yagnas” or “Yagyas”. There are not enough qualified priests in some of the countries and places where they promise these! It definitely sounds great and you visualizes a grand Vedic ceremony to wipe out all your sins. Instead of getting carried away, ask them > what all will be performed, how many recitals and homa offerings, by how many pundits it will be performed, the date time and the venue of the homa and as a final deterrent tell them that you or your representative with a family pundit would like to attend it.