Domo

"どうも (Domo)" has several meanings. 

1. Hi/ Hello/ Hey
2. Thank you
3. Somehow

If you use it without combining with other words and just say "どうも (Domo)", it usually means Hi/ Hello/ Hey and is a casual greeting between close friends. If you use it after someone does something for you, or when talking about something nice someone did for you, it's a short form of expressing your appreciation. (Kind of like the abbreviated version of "Domo Arigato Gozaimasu".) 

However, when you use it in within a sentence, it usually has this nuance of "somehow~" or "I'm not sure but~" or "it seems to be~", which shows the situation of something which is vague and not so clear. 

It's a little bit difficult to use "どうも (Domo)" in a sentence, so if you are a beginner of the Japanese language, I recommend for you to just use "どうも (Domo)" to a new Japanese friend when you want to greet him/her to simply say "Hi!".

Sumimasen/Suimasen

You may sometimes hear people saying "すみません (Sumimasen)" or "すいません (Suimasen)", in various situations. But what do these words mean?

The origin of the word comes from the word "済む (Sumu)" meaning end/ finish/ complete. The word "済む (Sumu)" also includes interpretations such as "it's solved", "~ is satisfied". This meaning changes when it is the negative form "～ません (~masen)" is added to the end of the word, and turns into "済みません (Sumimasen)", which the exact translation now means "can not be solved" or "can not be satisfied". 

Because the pronunciation of "すみません (Sumimasen)" gradually changed, it is pretty common for people to also use "すいません (Suimasen)" too. 

"すみません (Sumimasen)" and "すいません (Suimasen)" are used in various situations to express the following in a casual manner:

1. I'm sorry
2. Excuse me
3. Sorry to cause you trouble
4. Thank you

It's very useful because you can use it in various situations, and it makes you sound very Japanese by using this word, but please be careful, because it's basically a casual way of expression.

• When you bumped into someone on the train and want to apologize very casually.
• When you got lost in the streets and want to stop someone to ask for the direction to the nearest train station.
• When you want to call the waitress to request for the beverage menu.
• When you are trying on the size M clothing but it's too big for you, and the shop clerk brought the S size for you.

If you want to sincerely apologize or show your gratitude in a bigger way, it is better to use expressions such as "申し訳ございません(I apologize)" or "ありがとうございます(Arigato gozaimasu)".