Strike price (also called exercise price) is the price at which you can buy the underlying security when exercising a call option, or the price at which you can sell the underlying when exercising a put option. Spot price means the current market price.
In short: spot price = now, while strike price = when exercising.
Option Spot Price vs. Underlying Spot Price
In this context, spot price can actually mean either the current market price of the underlying security or the current market price of the option itself, which is why the use of this term with options can be a bit unfortunate and it's always better to say "option's spot price" or "stock's/underlying spot price" rather than just "spot price" to avoid any confusion, or "option's market price" and "underlying (market) price", or "option premium" and "underlying price".
Spot Price vs. Futures/Forward Price
The term spot price is not limited to options or stocks – you can use it when referring to the current market price of any security. It is most commonly used with securities which besides the spot market also have futures or forward markets, such as commodities, currencies or interest rates.
For instance, you can hear about the "gold spot price" as opposed to gold futures prices, or you can "buy euros on the spot market" as opposed to the forward market. Generally, spot price is the price for immediate delivery or settlement (in practice, immediate typically means settled within a very few, like 1-3, days), while a futures or forward price, although agreed now, is for settlement at a given date in the future (e.g. one month or even one year from now).
The key thing to remember is that in finance "spot" means "right now" or "immediately".
Example of Option Strike vs. Spot Price
Let's say you think that General Electric (GE) stock might go up in the near future and you want to buy the 30 strike call option which expires 3 months from now – an option which gives you the right (but not obligation) to buy GE shares for $30 at any time from now to the option's expiration. GE stock is currently trading in the stock market at $29.43. The 30 strike call option is currently trading at $0.75 per share in the options market.
Strike price = $30 = the price at which you would be buying GE shares if you exercise the option at some point. Whatever happens in the market, strike price with this particular option will always be $30, as it is fixed throughout an option's life.
Option's spot price = $0.75 = the price at which you can buy or sell the option itself (not the underlying stock) right now. Spot price only applies at this moment – tomorrow or in 5 minutes it can be different.
Stock's spot price = $29.43 = the price at which you can buy or sell GE stock in the stock market right now. Again, it can (and most likely will) change and be very different in the future.
You can see some more examples here.
Relationship between Strike and Underlying Price
The relationship between an option's strike price and the underlying stock's spot price (which of them is higher) determines "moneyness" of the option, which is another very important piece of option terminology: we say that an option is in the money (ITM), at the money (ATM) or out of the money (OTM). It is different for calls and puts.
ITM: underlying price > strike price
OTM: underlying price < strike price
For puts it's the opposite:
ITM: underlying price < strike price
OTM: underlying price > strike price
For both calls and puts:
ATM: underlying price = strike price
Generally, when you are long (you own) an option, you want it to be in the money, as that is when the option is most valuable (this is also clear when you think about the above listed relationships between strike and underlying price for calls and puts).
For more detailed explanation see ITM, ATM, OTM Options.