In standard Jewish tradition, I will try to answer you by asking another question.
There are two words that are used interchangeably, but really have very different meanings. What's the difference between "spiritual" and "holy"? Spirituality implies an attachment to matters of the spirit, rather than the physical, material world. A spiritual person is selfless, perhaps ascetic, one who strives to imbue his life with a higher meaning, to think of and serve others, not himself, to see the purpose and goodness in life. But spirituality is limited by one's own spirit, one's own perceptions, one's own self. An atheist can be a spiritual person. He can certainly be a good person, a wonderful person. He can go as far and as high as his own spirit moves him.
"Holy" refers to the divine, that which has its sanctity directly from G‑d or is connected with Him. When we do what G‑d tells us, we access holiness. We ascend higher than our own limitations; we connect to the essence of the Almighty Himself. We may not necessarily understand or relate to it, but when a man dons tefillin, or when a woman immerses in a mikvah, whether they feel spiritual or not, they are accessing G‑d. Eating on Shabbat is a mitzvah; conjugal relations (in the right time, under the right circumstances) are a mitzvah. They are holy. Holiness is not connected to asceticism; it is connected to G‑d.
Spirituality comes from man. Holiness comes from G‑d.
A practical example: a Jew may go to synagogue on Shabbat, pray, recite the kiddush over wine, etc. He may not feel particularly spiritual when doing these things, but in observing the Shabbat in the way that G‑d has asked, he is holy. Because he is connecting to G‑d as G‑d wishes. By the same token, he may decide to burn incense and play music on the Shabbat to enhance his spiritual feelings - but by violating the laws of Shabbat, he has strayed from holiness.
On the other hand, you may have a person who fulfills G‑d's mitzvot just as G‑d commanded him, but without any attempt at being spiritual, without love or fear of G‑d, remaining just as materialistic and coarse after the mitzvah as before. Judaism teaches that ultimately such mitzvot are soul-less, dead. The Torah commands us to "choose life." It is the spiritual striving to connect with G‑d that breathes life into the mitzvot.
So, what defines a person as religious? Ideally, a combination of both holiness and spirituality. A person who strives to fulfill G‑d's commandments - thereby being holy - but who strives to find the spirituality in what he's doing. A person who is not satisfied with rote performance of mitzvot, but who tries to connect with the Divine in all his actions. A person who, as part and parcel of his service of G‑d, places others, and "matters of the spirit," over himself and matters of the flesh.
Having said that, you must know that Judaism is first and foremost an action-oriented religion. So while both holiness and spirituality are vital, when push comes to shove we are enjoined to perform the mitzvot. Torah tells us, "Be holy" (Leviticus 19:2); it does not command us, "Be spiritual." Through being holy, ultimately we reach spirituality as well. Therefore - in a much narrower sense, a "religious person" is often defined as one who observes the three "biggies" in terms of holiness: Shabbat, kosher, and family purity.