There's no way to identify a miracle for sure. A miracle is something that happens due to some force that is outside of nature or of physical laws. But we don't understand these things perfectly enough to know when a miracle has occured. For example, if I get cancer and then it goes away, is it a miracle or is it simply that we don't yet understand the process by which cancer appears and disappears well enough?
A true miracle is something that involves divine intervention, which means God does something that impacts our mortal condition in some way. I don't know that I've ever experienced one, but I'm not sure I'd know if it happened. How can you be sure? You can't, I don't think.
The Catholics do have a definition of a miracle, and in order to be declared a miracle an event has to be investigated by Church officials. Commonly or colloquially, a miracle is something that cannot occur but does. Sometimes we use the religious definition and sometimes we just refer to a miracle as something unusual or unexpected.
A miracle is defined by the Catholic Church (and I don't see where their definition differs that much from other faiths) as "wonders performed by supernatural power as signs of some special mission or gift and explicitly ascribed to God." (see Catholic Encyclopedia at www.newadvent.org)
Miracles, then, are unnatural, wonderful occurrences and the hand of God can be clearly seen in them. I think, generally speaking, that people ascribe God's will to things that they can't otherwise understand, as in someone beating a disease, or getting a promotion at the time you need it most financially, or some other extraordinary event. I understand this, and will never take anyone to task on this belief, but it's not really one that I share. Wonderful things happen, but whether we identify them as miracles or not ultimately comes down to, like most things, personal faith. As I said, I certainly understand the need of some people to believe in them.