Why is the LM741 operational amplifier so prolific?
In 3 words - "Baby  Duck syndrome" - people who encountered the amplifier in a favourable environment when it made more sense, and who grew up with it, continue to have warm fuzzy feelings about it long after it is "past its use by date.

Once it made sense to use a 741 for many tasks and the habit has died hard. Some details below. It no longer makes sense but people keep on keeping on out of habit and familiarity.  Similar to the way that many projects specify a PIC 16F84 processor when there are a vast number of better and cheaper PIC processors now available. The PIC 16F84 is the microcontroller-mother-hen of a whole generation of engineers. 

Long ago (3+ decades) the LM741 was an extremely good choice.
 It rated "reasonably" in the areas  of bandwidth and slew rate, input offset, and noise. It was internally compensated (unlike the the current 709) and tended not to burst into glorious oscillation in the hands of the unaccustomed. It didn't latch up and/or die when you exceeded its input common mode range and it tended not to produce magic smoke when you overloaded its inputs or output. And it didn't cost too too much. For those who are used to opamps doing all theses things as of right the above list is a bit of a puzzle, but way back then the 741 tended to be one of the first "everyman's op amp"'s and tamed the issues that were typical among the higher performing examples of the day.

SO the 741 has earned a faithful following of people who remember it fondly.
 AND the people who mindlessly plagiarise and copy and steal  ideas and material from others keep on copying the 741 designs as well.

Note that a whole line of answers putting the other point of view may be found here

Arguably the biggest "problem" with the 741 is that it is not "single supply" - ie the input common mode range does not include or come near the negative rail, so it MUST be used with "ground" somewhere between the two power supply rails. This usually translates to needing dual power supplies, but a single supply of adequate voltage can be used with artificial reference ground being set somewhere between the supplies.

Output voltage swing is +/- 3V short of the rails with a 10 K load and +/- 5V Short of the rails with a 2k load. eg if you wanted to achieve a 5V total swing in a 2K load you need a minimum of 15 Volts supply voltage (or +/- 7.5V)

Input common mode range was 3V short of supply in each direction.