If you own individual stocks, bonds or other securities, you only have a gain when you sell the security for more than you paid for it. If you sell stocks, that gain will be taxed regardless of whether you cash out the money or reinvest it. But you don't owe anything if you just hold the stocks, even if they go up in value. You only have a gain when you sell and make the gain "real" ("realizing the gain"). The stock may also pay dividends during the year which are taxable when paid, even if you reinvest them in new shares.
If you own a mutual fund, then you may have a gain if the fund manager sells stocks within the fund, even if you don't sell your shares. This will be true whether the fund manager sends you the profit from the stock or reinvests it in other stocks. The stocks in the mutual fund may also pay dividends, which are taxable to you even if the fund manager reinvests them for you. However, these types of trade increases your basis in the investment so, while you may pay some gains tax each year, you will then pay less gains tax when you sell your shares.
All this information pertains to investments that are NOT part of a tax-advantaged retirement scheme (such as an IRA). If your investments are in a tax-advantaged retirement account, the rules for those accounts will apply instead.
