What is the Meaning of the Black-Scholes Value?

Black-Scholes is one of several pricing models that uses six variables to determine the theoretical value for an option. You mentioned five of them. You did not mention a dividend so it is assumed that there is none.

What's lacking in your question is an understanding of what a call option is, namely the right to buy an asset an agreed price on or before a particular date. If the option is European style (most indexes), they can only be exercised at expiration. American options (stocks, ETFs) can be exercised at any time.

In your example, the buyer of the call pays $234 (quoted at $2.34 per contract) for the right to buy the underying at $100, if so inclined. Should he prefer not to own it, he can sell the call at any time in the option market, perhaps for more, perhaps for less.

Puts are the mirror image of calls. The owner has the right to sell the underlying at the exercise/strike price, if so inclined.

In both cases, the owner of the option has the right. The seller has the obligation.

Using the data, the call option is $2.34, but what exactly does that mean? Does that mean the buyer has the obligation to buy the stock for $2.34?

No, a call option is when someone purchases the right to buy the stock at the exercise price. The obligation is on the seller to provide the stock. The buyer can just let the option go unexercised if the buyer does not want the stock at the exercise price. Many options are never exercised. This may mean that the buyer of the option no longer wants that stock. Or it may mean that the buyer can get a better price than 100. It only makes sense to use the option if the price is higher than 100.

The $2.34 is the price of the option itself. The buyer pays that immediately. Within the maturity period (two months in this example), the buyer of the option can call it and purchase at the exercise price (100 in this example). I.e. the 100 is the price at which the seller is obligated to sell the stock in this example.

The basic idea is that the buyer of the option wants to buy the stock if it goes up beyond 100. Black/Scholes/Merton determines a reasonable valuation of that option based on various parameters.