Red Flowers can be found in birch-like Ashenwood-trees growing on their levaes. These flowers are blocks that can be placed and even used as building-material. They glow very faintly, so they cannot be used as a light-source.
On top of the flower-blocks Leafis might rarely spawn, at night / in darkness also Night Leafies, even if the flowers were placed as a decoration inside a house made of crafted blocks. If placed into Flower Pots or Wood Planters, the flowers will not make any mobs spawn though.
Red Flowers can also be occasionally discovered in Wood Treasure Chest.
Ashenwood-trees can be grown from Ashenwood-Saplings (an additional harvest when gathering leaves), but only in fitting environments. Grassland is not always a good place to grow them. They can be grown on Grass-blocks with some Water not too far away (a few blocks, probably the water has to be in the same chunk 16x16x16 blocks large). The water can be a natural body of water or water that you placed yourself. Saplings can be fertilized by Pigsy Droppings, but only if the Saplings aren't fallow, because in this case no tree will grow from them at the chosen spot anyway.
More Red Flowers will - slowly - grow on Ashenwood-leaves with time, so you can lay out flat artificial biomes made from Ashenwood Leaves and wait for a few (real time) days until you can collect one or two Red Flowers there every couple of days or so. The same goes for Blue Flowers, Yellow Flower and Wildwood Flowers; not for Autumnwood Flowers nor Weepwood Flowers though.
A faster method to get more Red Flowers is to place blocks of Red Flowers on the ground and pick them up again - once in a while you will obtain 2 flowers at once when doing that.
Red Pigment is used for crafting Red Beacons, Red Lamps (Wood Burning Lamps emitting red light), Red Fireworks, Red Stained Glass, Red Beds, Orange Block of Goo, Red Concrete, Red Mosaic Tile Wall, Red Adobe Floor, Red Adobe Roof, Red Adobe Wall, Red Carpet and Red Wood Wall.
Red Flowers can also be used as an ingredient in Basic Health Potions, Stone Mining Cells, Taming Collars, Golden Potions and Stamina Regeneration Potions - in recipes where you can choose from different possible flowers.