Flying Blue is the airline loyalty programme of Air France, KLM, Air Europa, Kenya Airways, Aircalin, and TAROM. The intention of the programme is to encourage you to spend more money with these airlines [instead of with their competitors], in exchange for certain benefits.
SkyTeam is an airline alliance that includes all of the airlines above, plus several more. The point of the alliance is to offer you, the traveller, better connectivity when far away from your usual airline. For instance when you are travelling inside North America, you cannot travel on Air France. Instead you will probably travel on Delta, which is another SkyTeam member. You may earn miles into Flying Blue and have your Flying Blue status respected by other airlines in SkyTeam.
Flying Blue has a currency (air miles) which are accumulated on these airlines, and their partners. It also has a status system that entitles the member to various benefits, such as access to an airline lounge or fast track through security, when travelling on SkyTeam airlines.
Miles accrued into other programmes cannot be merged into Flying Blue, even if those programmes are also in SkyTeam. If you earn miles into Delta's programme, you cannot transfer them across. The programmes are separate. However, you can earn Flying Blue miles when travelling on Delta, and you can use your air miles to buy Delta flights, because Delta are in SkyTeam. You simply need to give your Flying Blue number to Delta when you buy your flight or when you check in.
You cannot buy flights directly from SkyTeam. They are a marketing organization only.
There are two other airline alliances, oneworld and Star Alliance. In general airlines tend to cooperate with fellow airlines in their alliance and compete with other airlines, but this is a rule of thumb and should not be applied too rigorously.