A paradox in Doctor Who is created when an action occurs that causes future actions to change in a way that makes the current actions impossible. There are tons of examples in the new series' that show when this does and doesn't occur. (An extensive description of Doctor Who paradoxes is found in my answer to this question.)
In Father's Day, when the "second" Rose saves her father from dying, she changes her own past in such a way that she likely never met The Doctor in the first place. Having two parents would have likely changed important factors such as where she lived and where she worked during the time when Rose (the episode) occurred. This is a paradox.
The second time a paradox is created is a lot harder to explain, but I suspect the word "paradox" is being used incorrectly here. The Doctor points out that having two of the same person in the same place "weakens" the fabric of time at that point, so having the touch would logically be even worse. By the time adult Rose touches infant Rose, the paradox is already in place; she has simply made things worse by weakening the already disrupted timeline, allowing the recent past events to "bleed through".
AFAIK we haven't seen anything else like this in future episodes; though there have been cases where other companions met their earlier/later selves, they've never made physical contact that I remember. There have been dozens of cases where different incarnations of The Doctor meet and interact, but in those cases there is usually some external force working to keep the timelines intact, or else (e.g. Time Crash) their meeting is even more catastrophic than the Rose's. The only other possibly-related event we see is when the sonic screwdriver touches an earlier/later version of itself, it always produces a nasty shower of sparks, and a remark from The Doctor about the "temporal disruption" it's causing.
Most other examples of paradoxes occurring in Doctor Who all seem to fit this basic pattern:
- In The Sound of Drums / The Last of the Time Lords, the paradox happens when future humans come back to kill past humans, who then couldn't exist in the future to come back and kill anyone, etc.
- In Angels Take Manhatten, the paradox happens when Rory kills himself after watching himself grow old and die as a prison of the Angels, meaning he never became the Angels prison and thus never had a reason to kill himself, etc.
Also note that time seems to be very "flexible" in this regard; there are many more cases of people changing past events, but having those events "repair" themselves, e.g. Waters of Mars, without creating a paradox. The key trigger seems to be when the newly-broken timeline becomes "unfixable" that a paradox happens.
