You may notice that when your seedling first comes up, the set of leaves that pops up looks nothing like the plant. Eventually those leaves start turning yellow or brown and fall off, which worries many folks who are new to growing seedlings. So why does this happen?
Two examples of cotyledons in cucumbers.
Those first leaves you see are cotyledons, whose function is to break through the dirt and feed the plant. Those leaves may stick around for days - or even weeks - depending on how fast or slow your plant grows. This is why seedlings can be grown in zipper bags, wet paper towels, peat, etc at the beginning - because the plant will feed off the cotyledons. Once they're spent, they will naturally wither and fall off the plant. So don't worry when they do so - as long as your plant has additional leaves, the plant will be fine.
The next leaves you see will be true leaves, which actually look like what the plant's leaves. These leaves will have no real nutrients on their own and will need to pull from the soil. If there isn't enough nutrients in the soil, your plant will struggle. At this point you may need to add fertilizer, depending on what kind of soil you used. This is especially true if you grew them in a soilless medium, like peat or many seedling soil mixes.
A common fertilizer at this stage is fish fertilizer. Just a warning - it does smell pretty bad. So be careful when you're pouring it on your plants not to spill it on yourself, your floor, etc. You'll use it at 1/4 the strength as normal, which is typically about 1/2 tablespoon per gallon of water. I typically mix it up in a gallon watering can, which makes the measuring pretty easy - you'll learn real quick about what size 'dollup' you need to pour into the watering can.
If it takes too long for the true leaves to pop up, this can cause the plant to use up its store of nutrients before it has a way to photosenthesize. If this happens, your seedlings can die. This is why it is so important to keep seedlings warm when they are growing (65-75° F). If they stall out for too long, they can become stunted and even die. A stunted seedling could take as long - or longer - to grow to full size and produce fruit than if you'd just started the seeds later outside.