Heathenry is a religion in which there is a long and complex tradition of gift-giving and receiving. This is clear from the Havamal, and from the study of runes such as Gebo. (Gebo is one of the reasons I recommend all Heathens spend some time looking at the runes, even if they never plan to use them. They have many lessons for us.) We give, and in turn we receive. If we receive, we offer thanks. We offer gifts to the gods, be they the food from our table, the mead from our horn, time, etc. These offerings, by the nature of the gods, are not and cannot be equal. But they are appreciated. The gods are friends, and are kin, as much as we serve them. Offerings take the form of sacrifice, and as sacrifices, they should be a sacrifice. You give what you have, in the sense that they won't ask something of you that's beyond your means. But don't offer water when you have ale - the gods are your guests if you invite them to your hof or hearth. Treat them with the best of hospitality.
Offerings, in the form of sharing one's own bounty and one's own meal or drink, are not simply offerings of thanks. They are a way in which we share of ourselves with the gods. When we pour to the gods we aren't usually giving something that we aren't partaking in ourselves. Taking a drink and offering the same to a deity strengthens bonds with that deity. It is a form of communing with them - a communion, I suppose - and a holy thing. When an offering is gladly accepted, that is a wonderful thing to feel - a marvellous, moving and sacred thing. Additionally, in a sense, sharing an offering with the gods is a renewing of contracts. Contracts in which we have declared to honour them, and they to aid us, or contracts in which we have made an oath, and so on.
If I'm contacting a god I don't know well, I'll usually feel obliged to give an offering as a thanks for listening to me. With gods I know well I'll pray to them without feeling that need, because we've already formed that relationship. Gods with whom I don't have as strong a bond, or one's I'm meeting for the first time or am barely acquainted with, I'll feel I should make an offering because in a sense I feel I am intruding upon their existence, and should make an offering like a guest bringing a bottle of wine to a dinner, or something.
I very much like the concept prayer, particularly repeated prayer. I have trouble finding ones I like that are easily repeated but I have written a few nice ones of my own. Most published prayers for Pagans are very... dull, something the worshipper should have been able to come up with himself. What I look for out of a repeated prayer is a good lilt and meter (The Lord's Prayer in English has a good meter to it, as an example), and meaning, but an ease and flow to it that means once the words are internalised the prayer can be repeated whilst the mind concentrates on communion with the divine. I enjoy reading beads, and the feeling of them in my hands. In practice, though, I'm more likely to speak from the heart. I consider all verbal, and some nonverbal, interactions with deity as a form of prayer, and much of my prayers consist of conversations of a sort even if they start out with something more formal. I'm also a fan of wordless prayer, wherein communication takes place via images, sensations, feelings, emotions.
I consider prayer to be an important aspect of deepening one's relationship with deities. I know a lot of people are rather stand-offish with deities, even within Paganism which has a tendency towards wanting, or claiming to want, that personal communion with deities. I wonder sometimes whether it's a sort of internal difference between lay-people and semi-priesthood... whether striving towards deepening that bond with individual deities is some sort of consequence or symptom of that sort of mindset or desire.