Arbitrary precision is sometimes useful for certain applications. In most cases however double precision is more than enough for most applications. The main problem is that it can't use the very efficient multiplication algorithms in the CPU, and have to resort to different algorithm with overhead. The WorkingPrecision does matter in performance:

a = RandomReal[{-10, 10}, {50, 50}, WorkingPrecision -> 6];
AbsoluteTiming[a.a;]
a = RandomReal[{-10, 10}, {50, 50}, WorkingPrecision -> 60];
AbsoluteTiming[a.a;]
a = RandomReal[{-10, 10}, {50, 50}, WorkingPrecision -> 600];
AbsoluteTiming[a.a;]
a = RandomReal[{-10, 10}, {50, 50}, WorkingPrecision -> 6000];
AbsoluteTiming[a.a;]

But only for really large amounts of digits...

You can use e.g.

NumberForm[%, 20]

to see all the digits. Or change the default in the preferences:

In the slightly highlighted field...

When you use:

a = RandomReal[{-10, 10}, {300, 300}, WorkingPrecision -> n];

for whatever n, the numbers that are generated are so-called arbitrary precision numbers. When you don't use a WorkingPrecision option:

a = RandomReal[{-10, 10}, {300, 300}];

it will create machineprecision numbers:

MachineNumberQ[a[[1,1]]]

Evaluates to True. While in the former case (independent of n) it is False.

Machine precision numbers do not track the precisions of numbers, while arbitrary precision does. This makes a HUGE difference! On most modern machine a MachinePrecision number will be a 'double precision floating point':

$MachinePrecision
15.9546  (53 Log10[2])

By default this will be used. Note that Mathematica only SHOWS 6 digits by default on screen! i.e. RandomReal[] gives 0.531239 (but in reality it has all 53 bits set randomly)...

Thank you for the reply. I was getting similar times for the example that you also showed but I did not know that Mathematica was only showing 6 digits instead of machine precision. How can I make displaying full digits?

From your response I can see that it does not matter 6 for 60 digits with WorkingPrecision there is a cost of keeping track and that is quite huge.

You are right. In Mathematica we deal with:

Precision[x] which gives the effective number of digits of precision in the number x,

and

Accuracy[x] which gives the effective number of digits to the right of the decimal point in the number x.

There is also:

MachinePrecision - a symbol used to indicate machine-number precision.

The last is somewhat appropriate to traditional interpretation.

I think being exact is not an easy concept. My problem, I want to work with double precision (at least) and when I am working with double precision I expect a similar performance from Mathematica that of MATLAB for simple (highly parallelizable) calculation like matrix multiplication.

a = RandomReal[{-10, 10}, {10000, 10000}];
AbsoluteTiming[f = a.a;]

this takes around 17 seconds. It is 6 digits, same as your example. With MATLAB double precision calculation takes 17 sec. 10.3.1 for Mac OS X x86 (64-bit) (December 9, 2015)

If you want to use more digits than 6 as in

a = RandomReal[{-10., 10.}, {300, 300},WorkingPrecision -> 16]

again time is 13 sec. That is the part is confusing me. Why even double precision (which is physical) does take that long.

that is an interesting observation, however, what you did corresponds to

a = RandomReal[{-10, 10}, {300, 300}];
AbsoluteTiming[f = N[a].N[a];]

just working with 6 digits not even double precision.