You've already had a bunch of good answers to this, from some very smart professionals, but I'll chuck my two cents in as well.
What is Blue Chips?
Does it means that the stock are able to operate profitably in good times and bad, and how do you identify one?
"Blue-chip" is a colloquialism - it refers to a stock that's big, reliable, profitable, usually a dividend-payer; the sort of stock that won't make you rich but it won't make you poor either. Think of stocks like Singtel, the big banks, the Jardines, that sort of thing.
Just because a stock is an STI component, though, doesn't mean it's immediately a blue-chip. Golden Agri and Olam are in the STI, but I don't think anyone would call them blue-chips.
Penny Stocks?
Penny Stocks is the same stock that we trade on the platform? Or does it operate separately on another platform
"Penny stock" is another colloquialism, and it refers to the opposite of blue-chips - shitty little companies that swim around in the dregs of the stock-markets, have incredibly volatile (or just straight-line-downward) share prices, and usually have very low share prices (so they look "cheap" to uneducated punters). They're still listed companies, so they trade on the same platform as non-shitty non-penny stocks, but they couldn't be more different.
These things are a minefield. Don't get involved.
Broker.
Why do some people trade using a broker, and not doing it themselves? From my research, some feel that broker are more experienced and are able to help out in exchange for some fees.
I don't know either. I mean, I get that some people (usually the olds, am I right?) would rather hear a human on a phone line saying "OK, you've bought 10,000 shares of Las Vegas Overleveraged Casino Concern at eleventy-three dollars and sixty-blurg cents a share", but paying
twice as much money for someone to help you trade instead of doing it yourself seems like a terrible idea. That's money coming out of your pocket. If you buy shares once a month, that's a few hundred dollars a year that you could spend on a nice holiday or a bottle of primo red or even more shares.
Seriously, don't be an old person. Use an electronic broker. Use Stanchart. They're the cheapest, and they're perfectly good.
Update: I forgot one thing that Sinkie mentioned. Due to a quirk of the Singaporean market, your broker (if that's an actual person) also takes on all the risk that you won't pay for your trade, or won't deliver the shares that you've sold. This is a great deal for the brokerages because they don't have to take on as much credit risk, but it seems like a distinctly shĂŻtty deal for the brokers.
I don't understand why Singaporean retail trading hasn't moved toward banks and brokerages taking the risk on themselves; you'd think a dedicated credit risk department would be better at these sorts of assessments than a dude who's juggling four phones and a Bloomberg terminal.
Broker; Fees
How does the fee works?
Each time you trade - whether it's a buy or a sell - you get charged a brokerage fee on the value of the trade. Typically this is around 0.2% of the market value, with a $18-$20 minimum; some brokers are more expensive (though more expensive brokers generally aren't any better), some are cheaper. Standard Chartered is notably cheaper because they don't put any minimum on the fee, so if you buy a couple thousand dollars worth with Stanchart you'll only pay about four bucks.