What do seeds need to succeed?
A Comfy Container - Almost any empty container at home can be used to start seeds - clean milk, yogurt, egg cartons, or seed starting trays. Any container must have drainage holes on the bottom. If you reuse containers each season, be sure to disinfect them with a 1:9 bleach solution to destroy any potential seedling pathogens.
Good soil - For best results, always start your seeds in a high quality seed starting mix (like Sloat Organic Potting Soil or E.B. Stone Seed Starting Mix). It will be light and fluffy to prevent compaction.
Consistent moisture - Seedlings must be kept moist (but not soggy) at all times. If they dry out just once, they are "toast"! Conversely, you don't want to drown them. Soil that is too saturated is an invitation for fungal problems.
Air Circulation - When starting seeds indoors, provide good air circulation (similar to wind fluctuations outside) to help prevent damping off fungus (causes stems to get spindly and fall over) and increase the strength of each seedling's stem. To do this, you can place a small oscillating fan a few feet away from seedlings.
Temperature - Most seeds will germinate just fine at room temperature. A heat mat is usually not necessary, but seeds that prefer very warm temperatures for germination (like tomatoes and peppers) may benefit from it. And most importantly ...
Let there be light! - When you start seeds indoors, do they get tall, skinny and fall over? The most common reason seeds don't develop into healthy plants is lack of sufficient light. If starting indoors, they must be in a sunny south-facing window that gets light most of the day or under fluorescent lights with 14 to 16 hours of light per day. (You should keep the lights 1-2 inches above the seedlings to prevent them from stretching towards it.)