Since the term dates back to the Mishnah (a code of law and instruction dating back roughly two millennia), many people have put forth definitions of the basic term as used in the Mishnah (For example, the Maharal). What we refer to here and this website is the Torah Im Derech Eretz as outlined by Hirsch. So let's start with one of his descriptions:
"The term Derekh Eretz includes all the situations arising from and dependent upon the circumstance that the earth is the place where the individual must live, fulfil his destiny and dwell together with others and that he must utilize resources and conditions provided on earth in order to live and to accomplish his purpose. Accordingly, the term Derekh Eretz is used primarily to refer to ways of earning a living, to the social order that prevails on earth, as well as to the mores and considerations of courtesy and propriety arising from social living and also to things pertinent to good breeding and general education." Pirkei Avos, Chapter Two, Mishnah Two, S. R. Hirsch
In this description, Hirsch is commenting on the Mishnah. The Mishnah says as follows:
רבי אלעזר בן עזריה אומר, אם אין תורה, אין דרך ארץ; אם אין דרך ארץ, אין תורה. אם אין חכמה, אין יראה; אם אין יראה, אין חכמה. אם אין דעת, אין בינה; אם אין בינה, אין דעת. אם אין קמח, אין תורה; אם אין תורה, אין קמח.
The translation of the relevant part is "if there is no Torah, there is no derech eretz and if there is no derech eretz, there is no Torah."
Hirsch's definition here is broad and seems to say basically that Torah observance occurs in the setting of life. It is not monastic. Accordingly, for Torah observance to be successful, a person's regular life must be in order too. By regular life I refer to what most people think of as a good person. In the good old days, say 50 years ago, people sought to be respectable, well-mannered, self-supporting, and educated. Today, good manners matter less, except sadly in the office. The same goes with education. We seek degrees in order to earn a living (if only for the credential). But people don't sit around the hearth reading Jane Eyre. Education for edification matters much less than it used to. Money rules today. It's hard to reconcile any religion with that. So I'm going to stick with the goals of yesteryear to continue this discussion. Hirsch was writing for 19th century Germany. In doing so, I believe he wrote for the 19th and 20th century Western world as well, particularly for America which is a Germanic country. I discuss that in my essay "The Yekke from Uman." The culture has changed, but not entirely.
Hirsch was a prolific writer, producing thousands of pages of brilliant prose. This includes a complete commentary on the Pentateuch (the first five books of the Bible), a monthly journal whose essays were reproduced in a multi-volume Collected Writings, and commentary on the Siddur and Tehillim, as well as other letters and essays. Only a portion of the writing addresses Torah Im Derech Eretz directly. Much of it is straight up discussion of the Bible, holidays, law, and various other topics. Hirsch had an original approach to much of this. His commentary on the Bible utilizes an etymological system of word roots and his explanation of commandments presents novel symbolism to show their meaning and benefits. It is extraordinary stuff.
There is more to Hirsch's program than is contained in his comments on the Mishnah, which as I said, are general. Particulars include the following:
1) A sense of connection to one's host society. While we maintain a separate Jewish identity, we also seek the welfare of our gentile neighbors. For example, he wrote:
“The more, indeed, Judaism comprises the whole of man and extends its declared mission to the salvation of the whole of mankind, the less it is possible to confine its outlook to the four cubits of a synagogue and the four walls of a study.” Samson Raphael Hirsch, Collected Writings, Vol. VI, “Religion Allied to Progress.”
2) An appreciation for quality secular learning. It is not all narishkite. For example, he wrote:
"The more the Jew is a Jew, the more universalist will his views and aspirations be, the less aloof will he be from anything that is noble and good, true and upright, in art or science, in culture or education; the more joyfully will he applaud whenever he sees truth and justice and peace and the ennoblement of man prevail and become dominant in human society." "Religion Allied to Progress" S. R. Hirsch
3) Good breeding and good manners. As we quoted above, "the mores and considerations of courtesy and propriety arising from social living and also to things pertinent to good breeding and general education."
4) A stress on performance of commandments and good deeds. With some people, study is everything. With Torah Im Derech Eretz, we are full dedicated and talk about all three. See the eighteenth chapter of his book 19 Letters for more on this.
5) Happy living. In his book Horeb, he talks about the importance of getting exercise. In his articles on the Sabbath, he talks about the simple pleasures of the day with family, food, and so forth. See the Collectd Writings for more on that. He visited the Alps and talked about their beauty.
6) Job training and earning a living. These are necessities for successful living. He wrote:
Our task in life has no greater enemy, and there is no greater cancer on our present state, than ignorance. Study Torah thoroughly—Torah, the Prophets, Ketuvim (Hagiographia), Talmud and decisors. And do not study out of a desire to be a rabbi. Study Torah as a businessman, a tradesman, an artist, a doctor, or a scientist. Collected Writings, vol. 7, pp. 157So hopefully, that gives you a sense of it. All kinds of people have written articles about R' Hirsch. I'm not trying to offer the definitive definition, just to give people a start on the topic.