How to fix the width of the table?

You can save horizontal space by either leaving out redundant information ("vs boxing") which shouldn't be in a table anyways, or by changing the table layout from a 2x7 to a 7x2 grid. The following code shows both solutions.

The first fits in a one column document, but will still be too wide in a two column document.

documentclass[]{article}

usepackage{siunitx}
usepackage{booktabs}

begin{document}
begin{table}
  centering
  caption[The examples of DTW distance for the Weizmann datasets.]
    {The examples of DTW distance for the Weizmann datasets. Activities shown vs
    boxing.}
  begin{tabular}{ccccccc} 
    toprule
    activity     & boxing & walking & running & jogging & waving & clapping 
    cmidrule(lr){1-7}
    DTW distance & 0.5796 & 1.7782  & 0.8616  & 1.1747  & 1.3516 & 0.969    
    bottomrule
  end{tabular}
end{table}
begin{table}
  centering
  caption{The examples of DTW distance for the Weizmann datasets.}
  begin{tabular}{l S[table-format=1.4]}
    toprule
    activity vs boxing & {DTW distance} 
    midrule
    boxing             & 0.5796 
    walking            & 1.7782 
    running            & 0.8616 
    jogging            & 1.1747 
    waving             & 1.3516 
    clapping           & 0.969  
    bottomrule
  end{tabular}
end{table}
end{document}

Results in a one-column document:

Results in a two-column document:

I totally agree with the answer of Skillmon (+1) but do not answer specifically to "how to fix the width", that is, how to force a table into the text width.

At this respect you have three possible approaches: the tabular* environment or the packages tabularx or tabulary. In the tree cases you must fix the width of the whole table to linewidth:

begin{tabular?}{linewidth}{<type pf columns}}

Where ? is *,x or y and the type of columns depend on type of environment. Since the difference between tabular* and tabularx is covered elsewhere, with tabulary could be:

begin{tabulary}{linewidth}{@{}LCCCCCC@{}}
....
end{tabulary}

Of course, the result will be still horrendous, because you cannot fit well 7 columns in about 8 cm (assuming a standard column) with so much text, so your options are:

1. Reduce the font. Maybe acceptable in other cases, but in your case will only fit (more or less) with tiny, and even then the header will need 3 or 4 lines (far from the elegance, imho).

2. Take the two columns for the table, using table*, or when the behavior of starred floats is a problem, using strip of the midfloat package (see here). Then the result is not too bad:

documentclass[twocolumn]{article}
usepackage{lipsum,array,xcolor,booktabs,tabulary}
begin{document}
begin{table*}
centering % tiny for table 
caption{The examples of DTW distance for the Weizmann datasets.}
begin{tabulary}{linewidth}{@{}LCCCCCC@{}} 
toprule
Two sequences & 
Boxing vs boxing & 
Walking vs boxing &
Running vs boxing & 
Jogging vs boxing & 
Handwaving vs boxing & 
Handclapping vs boxingcmidrule{1-7}
DTW distance & 
0.5796 & 
1.7782 & 
0.8616 & 
1.1747 & 
1.3516 & 
0.969bottomrule
end{tabulary}
end{table*}
lipsum[1-40]
end{document}

However, be able to fit in some way a table should never avoid ask yourself about the design. For instance, beside the redundant headers, really you need four decimals?