Before we get into the discussion, though, it might be helpful to explore what it means. What is 'showing'? What is 'telling?'.
Here's a quick example of telling.
Susan was angry.
This tells the reader how the character is feeling.
Here's the same bit of information, delivered with 'showing':
Susan's hands curled into fists.
By showing how a character is feeling, you will inspire empathy and interest by putting your reader into the moment.
It's not just about emotions and getting your reader to engage with your characters, though. You can deliver all kinds of information using 'showing'. For example, marital status can be shown by a character fiddling with their wedding ring, or a futuristic setting can be shown by a character swallowing a protein pill.
In this way,'showing' is linked to the craft of describing your story without bombarding your reader with dry information. If this is something you find difficult, try analysing the opening chapter of your favourite book and see how all the details of character and setting are incorporated. You might be surprised by how few details are used to evoke strong images and feelings and how often they are shown.
However, the advice to always 'show don't tell' is, frankly, ludicrous. Consider this example of 'telling:
On a cold November day in 1861, a baby girl was born.
Can you imagine how many words would be required to 'show' this bit of information? Lots. If we all stuck to the 'rule', every novel would be ten volumes long. Telling is, of course, a vital and valid method of writing. The trick is to work out when to use it.
A good rule of thumb is that you should 'show' the important stuff. Whether it's the emotional state of a character you really want your reader to empathise with, or an important (or exciting) event in your plot.
For example, let's say there is a significant change in the relationship of your characters and you currently deal with it with a one sentence description:
It was after he rescued her from the wild baboon that Judy realised she had met a man she could truly trust.
The description occurs after the event, so it is a missed opportunity for suspense (will he manage to save her from the baboon? Will it change her feelings for him?) and a missed opportunity for your reader to 'live' the emotional change alongside your character.
In this example, you might consider dramatising the event. By making it into a proper scene (and allowing your reader to experience the event along with your characters), you will deliver a stronger emotional punch.