However minimal your Hebrew education was, you learned three phrases: "Shabbat shalom" is the greeting for shabbes; "sheket b'vakasha" is the teacher's often futile plea for quiet; "todah rabbah" is the response to any nicety, that is, "thank you very much."
The word "modim" is the plural verb that comes from the same word as "todah." Perhaps the simplest way to translate it, then, is: [we] thank you. But if you look at English and other interpretations of this word, they often complicate it by adding the notion of "acknowledgment." Other translations use "proclaim," affirm," or some such idea. At its foundation, however, the process put into motion by the use of this word is one of appreciation.
What does it mean to thank? We tend to think of it in two ways. In day-to-day interactions with people, the act of expressing thanks, whether through words or a note, returns something to the individual who has given you something they controlled. It may be something material, or information you request or even just a moment of time. The act of generosity that is presumed to be selfless can be affirmed with "thank you" and potentially turned into resentment without it. In that sense, offering thanks is an investment in future acts of generosity toward yourself or others.
One the other hand, the notion of thanksgiving, which underpins the American holiday of the same name, is very much introspective. The act of offering thanks is designed to penetrate to the core of the individual who has taken stock of the blessings, spiritual and material, that are part of life. Giving thanks is a gesture of humility, and it articulates a critical part of successful human life: we are entitled to little if anything in this world. If the "attitude of gratitude" pervades our lives, we will be good stewards of the very life we live and circumstances that raise it above subsistence.
Of course, in prayer our words are addressed to God. Whether we are offering a thank-you note for our blessings or attempting to cure ourselves of taking things for granted, the introduction of God into the equation gives our prayers a different quality. We are not asking for anything; we are offering. We are not petitioners or supplicants when we thank; we are, instead, celebrants. And just as when we thank another person for a kindness, we affirm that a relationship exists. We affirm a connection.