Update - December 2022: As an example in this 2012 answer a then top of the range Sony APSC A77 camera was used as an example. While a lot has changed in the subsequent 10 years the answer is just as valid, or more so. Even in apparently "full manual" mode a modern camera (full frame or APSC) has many features which partially automate the photo taking task unless the photographer strives very diligently to disable them or to avoid using them.
Shooting in Manual mode and RAW probably comes closest to "true manual", but even then it is hard to avoid automatic features.
One example only - focus has become an even more "auto assisted" feature in the past 10 years.
Is focus truly manual? - is this maintained between frames or
is tracking used?
Eye focus? Animal eye focus?
With "manual focus" is focus acquistion indicated by the camera or is a split screen or brain judgement used?
Central spot focus or zone or multi region?
Animal eys focus?
Is ISO always set to one value by the photographer?
Is image stabilisation used in body and/or lens?
This is so common now that seeing it as "automatic" may seem strange - but, it allows a degree of camera assist that has utterly transformed the number of "keepers" that can be obtained 0 especially in low light.
"Auto" can mean a wide range of things.
Most DSLRs offer a "full auto" facility that tends to manage shutter speed, aperture, ISO and more. But most of the modes on a DSLR that are other than absolute manual mode offer a substantial automated component. And even "Manual" may have auto features lurking in the shadows (literally in some cases).
Your friends are correct to the extent that you will definitely be missing out on achieving the most that a camera system can offer if you use auto always — and certainly so if you are unable to use any other mode — but there are occasions when auto or something close to it makes sense for many people.
Some professionals use "auto" a large portion of the time. They would be in the minority.
Some use it occasionally, but seldom.
A hard core few would never use a full auto mode. You may well learn things from them but in most cases probably do not want to copy them — a Grand Master black belt 1st class may be wedded to the pure artistry of pure manual — but a top pro will use whatever tool does the job best/easiest/cheapest/quickest (choose some) and if that's full auto in some cases, so be it.
A postcard photographer with a tripod mounted camera, who sells photos to tourists may very much be a true professional (by definition) and if the modern magic toy's full auto mode invariably delivers the goods, then using full auto may be sensible.
A high profile web equipment reviewer and advice giver and shoots-photos-for-$ man (who I will not actually name) said some while ago that he always used auto mode as the cameras have got so good that they know best. I was very surprised, and this may not reflect his reality.
Most modes that a modern DSLR offer are semi-automatic, with some of the settings being adjusted by the system. If the camera can adjust at least something independently of the user's actions then it is at least partially "auto" - see below.
Aperture priority mode allows the user to manually set the aperture and then adjusts the shutter speed to suit the required exposure. ISO may be set by the user or the user may specify a range of acceptable ISO values, allowing the system to adjust ISO within this range. Some systems also allow the minimum auto-settable shutter speed to be controlled. Your don't-use-auto friends probably do not count all that as auto. It certainly is.
Similar applies to shutter priority where you set the speed and the system manages aperture and other settings.
"Manual mode" MAY allow dlighting or dynamic range optimisation to be carried out on a manual frame.
Sony's top APSC camera - the A77 (in 2012. In 2022 maybe A6400 or A6500), allows "dynamic range optimisation" in manual mode. (I just tried it to see). DRO is auto with a vengeance - complex signal processing is applied variably as required across the frame - that's "auto". The A77 (and later Sony APSC cameras) also allow in-camera 3 frame HDR — that's also auto.
I bought a Nikon D700 yesterday :-) (August 2012) ... quick check ... You can adjust the DLighting setting on manually setting frames after the photo is taken — suggesting that they are indeed applied at the time.
Personal comment: I'd perhaps be described as semi professional — if you don't count my time as worth anything then the obsession comfortably pays for the equipment. This does not make my wife too much happier :-). I take photos for my enjoyment and if others enjoy them too I'm pleased. If people enjoy what I do and how I do it enough to pay me to take them at an event etc for $ then I may agree. Works for all of us, so far.
So — I almost always use aperture priority mode — sometimes with auto ISO, but usually I prefer to have ISO under my control. It can be changed VERY rapidly on the A77. I also use memories for gross setting changes - which is also automation, even if the settings are manual ones, and then "play" around the memory produced settings. This allows eg rapid change during a wedding reception between ambient light no flash mode (large aperture, higher ISO, appropriate white balance) and flash photos (smaller aperture, lower ISO, white balance change.) Manually swapping between those memories is essentially an "auto" feature.
Very occasionally I try full-auto — usually "just to see what it knows" in a given situation.
I use AF largely but will drop into manual focus when apposite — either for one frame with AF/MF button or for a series of shots when needed. AF is "auto".
I usually use multi segment metering but may swap to center weighted or spot if appropriate. All those are "auto". The Sony Pellucid mirror cameras have the best live view in any DSLRs (the A77 utterly trounces my now 4 year old D700) and I frequently adjust lighting level on the fly with exposure compensation between shots based on the appearance of the live view screen. Sony make this extremely easy with control placement. Doing this is a MANUAL action standing on top of auto.
For critical focusing on the A77 I may swap to manual focus, then press the AF/MF button which autofocuses the camera, then release AF/MF so I am in true manual focus mode and use the focus magnifier to produce focusing as the lens is capable of - a superb capability. All that is a true manual MF mode sitting on top of an auto AF platform.
I'll use full manual mode on occasions when nothing else works. Occasionally things change so rapidly in a complex way that the brain is best at keeping up - and sometimes things change so little that manual is best. (Tripod and portraits.) Some years ago we did a large amount of driving in Europe. With several drivers in the car I was able to spend most of the time taking photos of the sights. We could not stop at all the places I would have liked to take photos of - this was a practical necessity in a most enjoyable situation. SO even when driving through eg a small German village that was not on our can-stop list I had to make the best I could of the situation. I had a top camera with the equivalent of a live view LCD AND an electronic EVF (and in real terms then cost more than the top ADSC DSLR now!). In the countryside I'd use aperture priority semi-auto. In villages I'd use full manual - aperture, shutter speed (and sometimes focus). Delve (optically) down a dark alley way - shutter speed to minimum safe, aperture to suit, zoom into alley, click, cafe table in sunlight - go on shutter up, aperture down, zoom back, click click, next ...
A mix of manual and auto used to advantage. You want to get to know your camera well enough to do this. Then use it on full full auto mode when it suits :-)