What does it mean to clone a gene?
Usually this is used in the context of gene analysis and further characterization. What is done, is that a piece of foreign DNA (either generated by digesting genomic DNA (from human or mouse for example) or by PCR amplification) digested at the end of the DNA to have matching open ends with a bacterial vector. 

The DNA of interest is then ligated with the vector and can then be replicated in bacteria and isolated easily in great yields. This plasmid can then further analysed (by restriction analysis, sequencing etc.), the gene or its promoter manipulated and mutated, or in case of protein coding genes the gene overexpressed in the cells. All this is useful to analyze the function of unknown genes. See the figure for a schematic overview (from  Campbell AP Biology Book 8th edition, via this website):



Technically making a clone means that you generate multiple genetically identical daughter cells from one precursor.