The basic evidence for global warming, that it is really happening, is that worldwide instrument measurements show that the world is warming and is continuing to warm. In fact, the last ten years were the warmest decade since instrumental records began in 1850, and the year 2010 was the warmest on record, slightly aboive the record set in 1998 and again in 2005, although because the temperature was only marginally higher, climate scientists regard the as the three equal hottest years globally.
It is difficult to find evidence to refute the instrument
measurements. The one serious doubt was based on the urban effect,
in which instruments located in originally rural areas gradually
become surrounded by urban development, resulting in localised
higher temperatures. It was suggested that climate scientists had
not properly adjusted for this effect where present, but
independent scientists have since confirmed that appropriate
adjustments have been made in the climate studies, and confirm that
global warming really is happening.
The evidence for global warming are the temperature records collected over the past century. These clearly show global temperature has increased, and dramatically so in the past 30 years.
There isn't any convincing evidence against global warming. There is paltry (a little bit) of evidence to suggest not ALL current warming can be attributed to human activity, but for the most part the evidence we have gathered so far implicates the human consumption of fossil fuels as the primary cause of current global warming.
If one focuses one's attention on a few specific areas of the globe rather than the globe in its entirety, one can build a flimsy case that global warming is not actually occurring. Some of those who argue there is no global warming pick one recent abnormally warm year, and draw a line from it through the coldest parts of more recent years in an effort to suggest temperature is declining, rather than using the standard five or ten year running average indicative of actual climate trend.