Otdfbt

The R language has a number of libraries to produce clean looking graphs, for example ggplot2.

The example below was made with the galluvial extension for ggplot2, one of many extensions that allow animation, interactive graphs, and more.

Slightly tangential: When it comes to the aesthetics of the output, the main advantage of the more technical tools recommended in other answers, over Excel and similar programs, is that they have better defaults. It is possible to create good-looking charts with Excel, LibreOffice, etc., if you know all of the things you need to tweak. In order to learn what those things are, I recommend these books:

• 
  The Visual Display of Quantitative Information, E. Tufte (the sequels are also useful, but not as much)


• 
  Data Visualization: A Practical Introduction, K. Healy


• 
  The Grammar of Graphics, L. Wilkinson


• 
  Preparing Scientific Illustrations, M. Briscoe (thanks @henning).

Reading these books will also help you get the most out of something like gnuplot or ggplot2.

LaTeX offers numerous packages, e.g., pgfplots.

Just to address your example specifically, the top example shows some signs of having been generated by MATLAB.

MATLAB has documentation about how to make such a plot here.

To add a little more completeness, if you know JavaScript and are looking to make a very unique visualization D3 is a nice choice that has a reasonably friendly API but gives the user very fine grained control.

Although it might possibly be true as some of the answers and comments state that good graphics can be produced in Excel, there is one very big downside to using Excel for scientific graphics. The major downside is that it is very easy to make mistakes in an Excel spreadsheet and very difficult to see that they have been made. Auditing a spreadsheet is exceptionally challenging - you basically have to try to reproduce the results by other means. There is a way round that problem but it involves adhering to extremely strict protocols as to the construction of the spreadsheet. Since most people produce complicated spreadsheets by starting simple and adding complications, the result will generally be unauditable (other than by reproduction).

ggplot is free to use and very easy to learn.

Excel can be used to create most graphics. It can be a bit click-y, but for one-off graphics, Excel is often the fastest way to create a custom graphic if you already know how to use it. The trick is to not accept Excel's defaults, so it doesn't look like out-of-the-box Excel junk that can be spotted from a mile away.

The one below was created in about 15 minutes (including making up data):

I'm a big fan of GraphPad Prism as I am not as skilled in the python, R or scripting in general (although I'm working on it). Prism is to me very intuitive and the GUI has all the buttons and taps for me to get the data to analyzed and then presented in a style very similar to the examples shown in the first post.