If you open a stock fund account with your Swedish bank, you would need to report some elements each year such as dividends paid, interest paid, and then any sales of the assets and the capital gains on them (or losses).
You may also be subject to requirements to file Form 8621 which relates to Passive Foreign Investment Companies ("PFIC"s). That form requires professional assistance for most taxpayers, but you won't be required to file it if your assets in PFICs is less than $25,000 (or $50,000 if you file a joint return with your spouse.) Some investments in PFICs are subject to fairly punitive tax, particularly if the reporting isn't managed from the beginning.
You may pay taxes on some parts of your investments to the US every year - but you would also be entitled to a credit for the taxes paid to Sweden. The reporting for investments from other countries differs, of course, from US reporting so you should keep good records of your investments and their values that you update regularly so that you can determine the necessary information for your US tax filing obligations. Exactly what your tax exposure will be depends on the types of investments that you make.
I would suggest that before or at the same time that you start investing you discuss this with someone knowledgeable in PFIC taxation, and ideally with some Swedish expertise as well. Please don't be discouraged from making these investments - just do so with some understanding of the reporting requirements.
For the FBAR, this would be a "deposit" account for "securities".
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