What are the differences and similarities among Paymium, Bitpay, Coinbase, etc.?
Paymium, operates Bitcoin-Central.net market exchange.  That exchange has partnered with a financial organization such that your EUR balance held at the exchange will be (once they go live with that) guaranteed by the government.  They will only be doing BTC/EUR when this goes live.  Currently they offer BTC/USD, BTC/GBP and BTC/EUR markets but no fiat balances are currently guaranteed.

BitPay is a payment processor for E-Commerce as well as for bricks and mortar / point-of-sale.  This allows a business to accept bitcoins for payment and the proceeds are delivered to the merchant as directed.  This could be 100% fiat (e.g., all bitcoin sales get converted to USDs immediately, or a mix, .. like 30% USD, 70% BTCs, etc.)   The currently can send payments to merchants in U.S., Mexico, Canada and several nations in Europe.

Coinbase account holders get a secure web-based bitcoin wallet, and by linking a U.S. bank account, the ability to buy/sell bitcoins directly from/to Coinbase at a USD conversion rate that is very close to the last price quoted from London based bitcoin exchange Bitstamp. Coinbase itself is not a crypto-currency exchange. Coinbase also offers bitcoin transaction processing services to merchants.

There are quite a few exchanges.  The bitcoin.it Wiki has Buying bitcoins and Selling bitcoins indexes that are fairly comprehensive as to the options available.