Genesis 24:1-9 states that "He will send his angel before you," suggesting humans receiving divine inspiration, but whenever this topic comes up I'm reminded of that classic scene in "Animal House" with the angel and devil on either side of the actor's head, making their respective arguments. Angels (like devils) are thoughts. As Hamlet stated, "There is nothing either good or bad, but thinking makes it so." Perhaps the artistic construct of a being with wings underscores the ethereal and speedy nature of thought, which could be considered just a form of energy.
Difficult question to answer. There is a verse in the Book of Hebrews in the Bible which says we should offer hospitality to everybody, because by so doing, "people have entertained angels unawares." I personally am not convinced about the whole heavenly creatures with fluffy wings, but I do think that we can be angels ourselves to each other, if that makes sense.
I have to agree with different aspects of each of the posts. While, historically, there is a definitive angel, I agree that there are angels here on Earth. People who go out of their way to help others without thought ring true of an angel. I agree that the term is sometimes overused, as stolperia points out. I honestly believe that for most, defining an angel is as different as the person him/herself.
The word "angel" has become overused in our secular world and has lost much of its significance. "Be an angel and bring me something to drink" is not an appropriate illustration of the intended meaning of the term! There are some who would argue that angels are or have been living among us - Mother Teresa, for example.
There is the traditional religious definition of angels, as described very well above, and there is also a philosophical and practical argument about the existence of angels and what forms they may take. Some believe that angels are the spirit form of those sanctified by God, sent to Earth to look over and protect the faithful. Others believe they are tied to the universe rather than to religion or God, per se, in the form of positive energy that can heal or effect positive change on Earth. So it is a very subjective, individual term that is usually used in a religiuos context, but not always.
This is not a theologically correct interpretation of the word, but to me, an angel is anyone who in some way brings God's word or God's grace to another. When we interact with such people, we are being brought closer to God. Who is to say that those people have not been moved by God to act in whatever way they did (the way that helped us). That is my personal view of what an angel is.
For an official Catholic view on angels, read this.
In Judaism, Angels are not necessarily divine in nature. The Hebrew word Malach, usually translated to Angel, actually means Messenger; before the standard depiction of Angels, God's messengers were usually ordinary people with divine missions. The divine nature of angels in Judaism is usually reserved for Named Angels, like Metatron, Michael, and Gabriel; these are, in effect, extensions of God's will: they have no free will or purpose beyond that which is given to them, and they cannot act on their own.
Interestingly, the concept of Satan in Judaism is not a Fallen Angel, but rather an abstract "Adversary," being the capacity for evil that exists solely in Man as a function of his Free Will.
People reinterpret angels throughout history. They are commonly considered to be people who were good but died and went to heaven. There are also archangels, which some people believe serve as guardians or can be called upon for help.
The Greek word "angelos" is the ordinary term for messenger. In Greek translations of the Old Testament and the original Greek of the New Testament, the term is used bith for ordinary messengers such as human letter-carriers and semi-divine beings.
Within Christianity, there is a hierarchy of being, in which the lowest kingdom is the mineral (which excels in durability), the next the plant (supreme ion taking in nourishment), the animal (excels in sensing), the human (excels in reason) , the angelic (excels in adoration) and God.
There are several hierarchies of angels elaborated in Christian theology, with various duties ranging from adoration of God through carrying messages from God to humanity to looking over humans.