When practicing mindfulness we choose to be aware of our flow of experience or of one aspect of our experience (breathing, for instance) with acceptance. What does "with acceptance" mean?
It means that we note the impact of the experience on our senses and our thoughts without denial, without turning away and without engaging in a mental drama about it.
Suppose I have agreed to give a presentation to work colleagues next week. Suppose also that I notice a ball of tension in my stomach whenever I think about the presentation.
Acceptance means I choose to experience that ball of anxiety without getting into a drama (OMG what if my boss is there, what if I stumble over my words, what if everyone notices I'm blushing?), without denial and without turning away.
In acceptance I refuse to waste energy fighting with that ball of anxiety. Instead I allow myself to experience my anxiety while I do what needs to be done, namely to prepare my presentation.
Usually, the acceptance in itself reduces the anxiety. And when I go on to do the preparation, the anxiety will usually reduce further. Sometimes the anxiety stays - but acceptance means I am not adding to it through generating pointless fearful thoughts that make matters worse while achieving nothing.