R-squared, is it an explanation of variation or variance? In most of the articles, it is written r-squared is the proportion of variance explained,however, in some articles, it is written as the proportion of total variation explained by the regression. Now, I am pretty confused, actually which one is correct and which one should I write in my thesis?
Variation is a general term which express the dispersion. Variation is not precisely defined, it talks only about the quality of some process which produces imprecise results and express the imprecision. Variance is a measure for variation, which is defined as the second central moment. There are many measures of variation like range, standard deviation and so on. However in the context of r squared variation could be only variance.
[Later edit to address the comment]
You mean that actually we measure variation by using variance like we measure weight by using kilogram or ounce or pound.If I am wrong please correct me.
Well, it's not quite right. The kilograms, ounce or pounds are measurement units, all of them have a precise meaning which is the weight. Staying closer to your example I would say that variance is to variation like the weight of your object is to how big an object is. In this case an object can be big because it has many kilograms, a big volume, a large surface area, an so on. Notice that in "how big an object is?" the notion of big is not precise, and can be understand slightly different, depending on what you are interested in.
Going back to variation, the point is that you can interpret that variation in different ways. Consider a sample which comes from a distribution. The variation of the values from the sample is talking about how the values are spread. Depending on your context the characteristics of that spreading which interests you can be different. You might use variance because you perhaps suppose a normal distribution and you want to compare with a statistical test if two samples have a significantly different mean. You also can use their range (maximum value minus minimum value) because perhaps you want to display them on a screen. Also you might use variance to compute R-squared for linear models to have an idea about how well the model fits the data. Obviously variance and standard deviation are prevalent due to their statistical meaning which helps in conducting inferences.