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What are the differences among these three ways of transferring money? Which one should I choose for transferring small amounts of money from the US to other countries, from an individual to another? Which one is the cheapest?
ACH
Wire transfer
Swift
ACH
ACH (Automated Clearing House) is a general name for a batch clearing house generally used for checks clearing. It is also used for transfers between the accounts that are made through, essentially, electronically generated checks. As such, it takes days for the transfer to go through (depending on how many clearing houses are there on the route, as each has a "daily" batch process). Typically transfers clear within a couple of days since all the clearing houses are interconnected and many (most?) banks end up using the same clearing houses (in which case it is done within a single daily process).
ACH transfers are based on electronic/actual checks and can be disputed/reversed.
Wire transfer
Wire transfer is a generic term for online transfer. As opposed to ACH transfers which are batched and aggregated daily and banks only move the differences between the transaction totals, wire transfers are processed online as they occur, or are batched frequently throughout the day. As the result, wire transfers typically arrive within hours (may take longer across borders or when small banks/credit unions are involved).
Wire transfer is usually more expensive, and generally irreversible. However, you can safely wire any amount to any bank in the world and it will arrive there within hours.
Swift
SWIFT is an international network connecting most (all?) the financial institutions in the world. Many international wire transfers are processed through this network.
There are other networks, for example within the US wire transfers are processed through the Federal Reserve system, and in Europe and some additional countries in the Middle East and Asia, the IBAN system is used for wire transfers.
Other examples for wire transfer networks include money transfer services (MoneyGram, Western Union, etc).
Additional methods to transfer include various "crowd-sourced" systems, like Hawala. In general terms, this is what was described to you in the other (spammy) answer that was trying to sell you this service. This is illegal in the United States and many other countries, if not done with proper licensing and registration. Generally, avoid suggestions to transfer money in avoidance of the financial reporting rules (aka: bitcoins, hawala, move concealed cash across borders, etc). Ideology aside, this may land you in a lot of trouble and cause significant inconvenience and money loss.
Which one should I choose for transferring small amounts of money from the US to other countries, from an individual to another?
Of the methods you mention, ACH is US only and the other two are basically equivalent to most US banks. You ask for a "international wire transfer" and they will arrange for it via SWIFT. That said, any possible differences between the two should be explained on your bank's own website if they treat them differently.
Which one is the cheapest?
As mentioned above, for many banks there isn't a difference between wire & SWIFT for an international transfer. Even if there is, no one could quote you cost recommendations without knowing your particular bank as the fees very widely between banks, and even vary widely between different customer classifications at banks. For example, "premier" customers will pay less than "standard" customers.
If your question was about inter-US transfers, then ACH will definitely be cheaper (in fact, free) but many banks don't allow ACH transfers between you and another legal entity. That is, unless they call it something like a Person-to-Person payment rather than an ACH transfer.
