Pulsars were a serendipitous experimental discovery made in 1968 by Jocelyn Bell and Anthony Hewish. They were discovered using a radio telescope array just outside Cambridge.
Anthony Hewish was working on a method for identifying quasars through a discovery in the mid 1960s that quasars scintillate (fluctuate in detected intensity of their radio emissions) more than less compact radio sources. He designed a large radio telescope to do this, which still remains in a field a few miles outside of Cambridge, as part of the Mullard Radio Astronomy Observatory.
The 'telescope' doesn't look like what we now consider a traditional radio telescope; it consisted of about 1000 posts stuck into the ground in neat rows with over a hundred miles of wire stretched across them to create about 2000 dipoles. The telescope scanned across the sky with time due to the rotation of the Earth.
Jocelyn Bell began her Ph.D. with Hewish at this time. Her task was to operate the telescope and analyse all the data, with Hewish as her supervisor. The telescope produced pen-and-chart data in huge quantities, all of which had to be analysed by hand.
Bell could immediately detect lots of earth-based interference (a permanent problem with radio astronomy) and also identified many scintillating sources. However after operating it for a while, Jocelyn Bell detected what she later describes as 'scruff' - neither interference nor a scintillating source. It disappeared for a while then came back, and she captured it on the chart as a series of pulses, which proved to be equally spaced corresponding to a frequency of about 3 seconds. Hewish believed them to be man-made, as with a gap of just 3 seconds the pulses seemed too close to be natural. Jocelyn Bell believed that it was possible that they originated from stars rather than man-made interference, so they investigated further.
They ruled out potential sources of man-made interference one-by-one. A different group of researchers then managed to pick-up the same signals, removing instrument error as a source. As the puzzle deepened, they considered the possibility that this regular signal coming from so far away could actually be some form of life in the far-distant universe.
While continuing with her actual Ph.D. research, Bell identified a second piece of 'scruff' close to Cassiopea A (itself a supernova remnant) and managed to capture the regular pulses about 1 second apart. This significantly reduced the possibility of distant life, and Bell went back through the miles of chart data that she had accumulated looking for more 'scruff'. She identified two more lots of 'scruff' and several other potential anomilies.
These additional discoveries confirmed to Hewish and Bell that this was neither man-made interference, nor was it (probably) alien life, but was some form of emission from these stars. Hewish and Bell wrote up their results and had a paper published in Nature. It sparked massive press interest (particularly the part about other life forms in the universe!) and so the discovery of pulsars was made. This is widely recognised as one of the great astronomical discoveries of the century.
Anthony Hewish was awarded the Nobel Prize for Physics in 1974 for the discovery of pulsars, along with Martin Ryle (also of Cambridge) for his work with radiotelescopes and particularly aperture synthesis. Sadly Jocelyn Bell, despite her instrumental role in the discovery of pulsars, did not share the prize.