I want to buy gold but I do not want to physically store it. I just want to invest in the price of gold. Where can I buy electronic gold certificates? Something like putting money in the bank but instead of dollars, I'm depositing gold, for example.
You could buy shares of an Exchange-Traded Fund (ETF) based on the price of gold, like GLD, IAU, or SGOL. You can invest in this fund through almost any brokerage firm, e.g. Fidelity, Etrade, Scotttrade, TD Ameritrade, Charles Schwab, ShareBuilder, etc.
Keep in mind that you'll still have to pay a commission and fees when purchasing an ETF, but it will almost certainly be less than paying the markup or storage fees of buying the physical commodity directly.
An ETF trades exactly like a stock, on an exchange, with a ticker symbol as noted above. The commission will apply the same as any stock trade, and the price will reflect some fraction of an ounce of gold, for the GLD, it started as .1oz, but fees have been applied over the years, so it's a bit less.
You could also invest in PHYS, which is a closed-end mutual fund that allows investors to trade their shares for 400-ounce gold bars. However, because the fund is closed-end, it may trade at a significant premium or discount compared to the actual price of gold for supply and demand reasons.
Also, keep in mind that investing in gold will never be the same as depositing your money in the bank. In the United States, money stored in a bank is FDIC-insured up to $250,000, and there are several banks or financial institutions that deposit money in multiple banks to double or triple the effective insurance limit (Fidelity has an account like this, for example). If you invest in gold and the price plunges, you're left with the fair market value of that gold, not your original deposit.
Yes, you're hoping the price of your gold investment will increase to at least match inflation, but you're hoping, i.e. speculating, which isn't the same as depositing your money in an insured bank account. If you want to speculate and invest in something with the hope of outpacing inflation, you're likely better off investing in a low-cost index fund of inflation-protected securities (or the S&P500, over the long term) rather than gold.
Just to be clear, I'm using the laymen's definition of a speculator, which is someone who engages
in risky financial transactions in an attempt to profit from short or medium term fluctuations
This is similar to the definition used in some markets, e.g. futures, but in many cases, economists and places like the CFTC define speculators as anyone who doesn't have a position in the underlying security. For example, a farmer selling corn futures is a hedger, while the trading firm purchasing the contracts is a speculator. The trading firm doesn't necessarily have to be actively trading the contract in the short-run; they merely have no position in the underlying commodity.
In addition to the possibility of buying gold ETFs or tradable certificates, there are also firms specializing in providing "bank accounts" of sorts which are denominated in units of weight of precious metal.
While these usually charge some fees, they do meet your criteria of being able to buy and sell precious metals without needing to store them yourself; also, these fees are likely lower than similar storage arranged by yourself. Depending on the specifics, they may also make buying small amounts practical (buying small amounts of physical precious metals usually comes with a large mark-up over the spot price, sometimes to the tune of a 50% or so immediate loss if you buy and then immediately sell).
Do note that, as pointed out by John Bensin, buying gold gets you an amount of metal, the local currency value of which will vary over time, sometimes wildly, so it is not the same thing as depositing the original amount of money in a bank account. Since 2006, the price of an ounce (about 31.1 grams) of gold has gone from under $500 US to over $1800 US to under $1100 US. Few other investment classes are anywhere near this volatile.
If you are interested in this type of service, you might want to check out BitGold (not the same thing at all as Bitcoin) or GoldMoney. (I am not affiliated with either.) Make sure to do your research thoroughly as these may or may not be covered by the same regulations as regular banks, particularly if you choose a company based outside of or a storage location outside of your own country.