This is a great question, for many reasons. In essence, you are asking the question of "What is life?" and then "How do we detect that life?"If we keep our scope fairly narrow here, we can look at some very basic principles of what it means for a cell to be "alive." Think about this for a minute. You can even ask yourself what it means for your entire body to be "living."

Let's start with a few basics.
How about "breathing?" Do cells breathe? Not in the way you might think about your lungs working, but they do "exchange gas" (like oxygen).

Hmmm... do they "eat" (or "consume energy")? They sure do! Your cells have metabolic enzymes that break down proteins, fats and sugars into energy packets that can be used to build and regulate the cells.

Another key aspect of being "alive" is being able to reproduce. Do cells do that? Yep, most (but not all) of them do, especially the ones that make up your skin, your hair and the lining of your gut. They undergo cell division (a process called mitosis).

Finally, cells that are alive and healthy are able to maintain their structural integrity - they do not have rips in their outer membranes and their nuclei and other intracellular organelles are also membrane bound.

Now, how do we detect these things in cells? It turns out that there are lots of ways to do so. If a cell is not alive, it breaks down structurally and we can detect this pretty easily under a microscope. There are many biochemical ways to detect gas exchange and metabolic processes. And finally, we can detect cell division microscopically as well. Some types of human cells can be kept alive in a culture dish, at least for a while, if provided with the right nutrients and conditions (as well as proper temperature). Looking at cells in culture (in vitro) helps us understand how they work.

Um, well, because if your cells weren't alive, then YOU wouldn't be either! Cells have to be living in order to perform functions; dead muscle cells don't contract, dead nerve cells don't carry information, dead red blood cells don't carry oxygen (and you know this if you're faint, short of breath, etc,) etc. Cells involved in digestion may be less obvious, but generally speaking, if your cells aren't living, then they don't function, and pretty soon the rest of you won't be living either.

It makes you think about what it means to be alive, doesn't it? Some things that living things do are to take in energy, give off wastes, trade gasses (like oxygen and carbon dioxide) with the environment, and reproduce.

It's difficult to see these things going on because each cell is so small. , but our bodies are made of trillions of cells doing these things. Our blood has to carry oxygen to each cell and carry off carbon dioxide so that our cells can turn food into a type of energy they can use.

A better piece of evidence is that some of our cells are always reproducing. Think of all the cells you lose every day. They get scraped off your skin and the inside of your digestive system. They die defending you from disease, but they are replaced when your cells divide by reproducing. Not all of our cells reproduce, but most do. If we were made of non-living parts, like a car, our tiny parts couldn't replace themselves and we'd be getting smaller every time we lost one.

What else separates living things from non-living things? Is fire alive?

There are a number of ways in which we can tell which cells in our body are alive. The first of which is that they undergo metabolism, which involves the intake of raw materials into the cell and then chemically converting these materials using enzymes to release energy or to produce other useful substances that allow the cell to function. The products of these metabolic pathways are crucial to maintain a regulated internal environment in the cell (aka homeostasis). Cells that are unable to maintain homeostasis will ultimately die. These products are also used in two other important fashions: for cell growth, and cell reproduction. Live cells are able grown by increasing in size, and sometimes if they achieve sufficient size will reproduce by dividing, producing two new cells.