A basic transcription factor consists of a DNA binding domain and usually another activating domain. Both domains are very important for transcription activation. Some transcription factor may contains a third domain, such protein/protein binding domain, that allows the factor forms homo- or hetero- dimer to regulate different downstream gene expression.
Only DNA-binding is enough for "transcription factor activity". If a protein cannot bind DNA is no longer a TF. As Chen mention, it is very common the presence of other domains like activators. In those cases, we should talk about "transactivation activity".