What are the differences between vipassana and mindfulness meditation? I have been reading about vipassana and mindfulness meditation but so far I haven’t fully understood the differences. Is mindfulness meditation an aspect or a variation of vipassana meditation?
When Buddha taught meditation he did not explicitly separate it in different types. If you read Anapanasati Sutta, Satipatthana Sutta, Kimattha Sutta, Cula-suññata Sutta, and any number of suttas mentioning the Jhanas, you will see that the overall progression is to first learn to pacify the mind at will, then learn to gladden the mind at will, then develop deep quiet satisfaction, then deep non-judgmental awareness, then turn the mind to seeing the implications of how things work (impermanence, corelessness, dukkha etc.), and then turn the mind to liberation.

The earlier stages of this process are referred to as Samatha and the later stages -- as Vipassana (or Shamatha and Vipashyana in Sanskrit).


Because the goal of samatha is to "tame" the mind, samatha is designed to be rather invasive. One target object is picked (a certain physical or a visualized object - depending on the nature of disturbances dominating the practitioner's mind) - and then a one-pointed concentration on that one object is cultivated. This requires mindfulness aka non-distraction: the ability to hold the object of concentration firmly without forgetting what you're supposed to be doing.
Once the mind gets minimally pliable and you can hold attention on one object for several minutes, you can switch to anapana-sati (mindfulness of breathing) meditation. This is when you use your breathing as a feedback to gauge the state of your mind and calm the mind. This type is what's usually called "mindfulness meditation" in the West. As you cultivate this, you develop a type of concentration that gives you direct access to the overall state of your mind, what used to be largely subconscious. Some people call this vipassana, but in my opinion this is somewhere in the middle between samatha and vipassana.
Once you learn to calm the mind using the breath as an indicator, you switch to gladdening the mind using specially constructed narratives. Basically, instead of telling yourself a sad story in which "how it is" and "how it supposed to be" mismatch, you tell yourself a joyful story in which "how it is" and "how it supposed to be" match gloriously. As you cultivate this, you learn the functioning of dukkha/sukha in context of Second and Third Noble Truths. These are already first elements of vipassana. At this point the object of your meditation is not as much breathing, as the thoughts in and off themselves - as explained in the Satipatthana Sutta. 
Once you learn to gladden the mind using the narratives, you switch to generating joy/pride directly by consolidating the mind around experience of surety and rightness. At this point the object of your meditation is not as much thoughts, as the mind in-and-of itself - as explained in the Satipatthana Sutta.
The next level is to get over the generation of joy and cultivate the quiet state of satisfaction, grounded in the experience of inherent purity and harmony. There is still a little bit of samatha in this.
The next level after cultivation of satisfaction is cultivation of pure awareness. Just sit and watch whatever happens inside and outside without making any qualitative assessments of the experiences. Don't go with thoughts, don't go against the thoughts. We could call this 100% mindfulness meditation and 100% vipassana. This is the level at which the Heart of Prajnaparamita Sutra should be studied.
The next level is pondering the real-life implications of the core Buddhist principles: impermanence, corelessness, dukkha etc. while keeping the level of calmness and concentration achieved on the previous levels. In my opinion, this is no longer mindfulness meditation, but is still 100% vipassana. This is the level at which the Realization happens.
The next level is what's known as meditation of non-meditation, or cultivation of non-cultivation: experiencing uncontrived natural mind. This is no longer vipassana nor mindfulness. You go beyond abiding in any station and enter what's known as the Diamond Samadhi.


So, as you can see it is hard to say what is mindfulness and what is vipassana. Most of the time things are not so cut and dry:

Some Tibetan schools prefer to emphasize samatha meditation, and require their students to master a complete one-pointed samadhi-with-seed before proceeding straight to the pondering meditation - something I personally found very frustrating... In this case the watching / tuning one's mind phase is implicitly included in samatha.

The Theravada schools seem to teach very strict vipassana as a watertight attention radically focused on microscopic level of phenomenological experiences. IMHO this is very prone to having some practitioners miss the forest behind the trees...

Some Zen schools have their students practice all of the above all at once, without separating it into any levels or giving any hints about the method or the goal. The idea is that a capable student will figure out the path by intuition. Not everyone does. You could call this kind of undifferentiated watch-and-figure-out meditation as "mindfulness meditation"...

Other Zen schools (like Thich Nhat Hanh's) actually teach the whole progression step-by-step: calming, gladdening and all -- and call that "mindfulness meditation"...

This whole Buddhism thing is a mess. People call things however they want. If you understand what is actually going on, like the levels above, you will not be fooled by names anymore and your practice can proceed without confusion.