The Icelandic words sem and að can both translate as that in English, which makes it easy to get them mixed up. Let’s make it a little easier to separate the two.

Sem and að have a few different meanings, but where people tend to confuse them (and what we’ll be talking about here) is sem as a relative conjunction (the English equivalents, who, whom, that, and which, are usually categorised as relative pronouns) and að as a subordinate conjunction. Basically, here we’re talking about sem and að when they connect things.

Rule of Thumb

The difference between the two words is quite simple. Basically, it’s like this:

sem follows nouns, while að follows verbs

Yes, this is a simplification, of course, but it’s a very useful one for now so just, like, don’t question it, okay? Both words can serve in different roles, but nobody mixes them up in those roles, so this is the difference that matters to us now.

Að halda (to think), að vita (to know), and að efa (to doubt) are all verbs, and thus they’re followed by að, not sem.

• Ég held að jörðin sé flöt > I think that the earth is flat.
• Ég veit að þú ert fáviti > I know that you are an idiot.
• Ég efa að þetta samtal skili neinu > I doubt that this conversation yields any results.

Konan (the woman), strákurinn (the boy), borðið (the table) and Myanmar are all nouns, and thus they’re followed by sem, not að. 

• Konan sem uppgötvaði radium hét Marie Curie > The woman who discovered radium was named Marie Curie.
• Strákurinn sem ég talaði við á barnum var víst giftur > The boy whom I talked to at the bar turned out to be married.
• Borðið sem ég smíðaði brotnaði undan þyngdinni > The table that I built broke under the weight.
• Myanmar, sem er í Asíu, ekki Afríku, er miklu stærra en Ísland > Myanmar, which is in Asia, not Africa, is much bigger than Iceland.

As a side-note, isn’t it nice that Icelandic is super simple with just one word (sem) to English’s four (who, whom, that, which)?

It’s Weird to Drop Words, and You Should Stop Doing It

Note that you cannot omit sem or að like you can that in English.

• Ég held að þú sért drullusokkur > I think that you are an asshole.
• Ég held að þú sért drullusokkur > I think you are an asshole.
• Þetta er borðið sem ég smíðaði > This is the table that I built.
• Þetta er borðið sem ég smíðaði > This is the table I built.
  

What you can do, though, is insert an að after sem. This is not unique to sem; you can do this after virtually any conjunction. It doesn’t change anything, but it’s considered non-standard. Avoid it in formal texts.

• Þetta er borðið sem ég smíðaði > This is the table that I built.
• Þetta er borðið sem að ég smíðaði > This is the table that I built.