Samatha meditation is more about calming your mind. You focus directly on one thing, usually your breath, and just keep your concentration at that. You don't necessarily ignore other things but you keep bringing your concentration back to your breath. As your concentration improves, you can develop a real single-pointedness of mind where fewer distractions and such arise.
This is a great time to start vipassana. In vipassana, your entire body and mind is the subject. You keep your attention on whatever is more prominent; usually that is your breath, but if you have an itch or a thought or a feeling, move away from the breath and focus your mindful attention on that instead. If you're thinking, say to yourself "thinking, thinking, thinking" until the thought ceases, or if you feel a breeze, note "cool, cool, cool" with your mindful attention until it stops. That's the essence of vipassana, realizing that all experience, whether those experiences are physical like heat or an itch, or mental, like an emotion or a thought, are impermanent and not under your control.
So basically samatha is more for relaxing and calming the mind through pure focus on a single object, mainly the breath. Vipassana is mindfully examining all experiences, whether physical or mental, and seeing them as impermanent and non-self. During vipassana, the breath is almost like a base camp, where you return to if there is no other experience that comes to the forefront. So explore and examine your thoughts and sensations, but when they calm down, just return to the breath!
I hope that helps. PM me if anything was unclear. Also anyone who sees any mistake or misstatement I may have made, please correct me!
Gain and loss, status and disgrace,
censure and praise, pleasure and pain:
these conditions among human beings are inconstant,
impermanent, subject to change.
Knowing this, the wise person, mindful,
ponders these changing conditions.
Desirable things don’t charm the mind,
undesirable ones bring no resistance.
His welcoming and rebelling are scattered,
gone to their end,
do not exist.
- Lokavipatti SuttaStuff I write about things.