You shall not murder
This law is applicable to modern Christians because it was well established before the 10 commandments (Gen 4:10-11, 9:6) and is continued into the New Testament (Matt 5:21-26, Romans 13:9).
This is a law against the willful taking of life, and perhaps also against negligence or carelessness resulting in death. For the modern Christian, this is a law against disregarding God’s design for living in a peaceful society (Romans 12:18).This is a law against rejecting the rule of the creator God by claiming authority over life that only God rightly has.
But perhaps most significantly, it is a law against an attitude of the heart that does not love others (See Matt 5:21-26). That is, if ‘you shall not murder’ is the negative command, the New Testament clearly makes the positive version of the same command to be ‘love your neighbour as yourself’ (Romans 13:9). For modern Christians, the law to not murder is trumped and fulfilled in Jesus’ teaching to love others (Matt 7:12).
You shall not steal
This is a law against taking what is not yours. This law has a horizontal aspect of not taking from other people, but also a vertical aspect of not taking for yoursself things that God hasn’t given you. That is, it is a sin of rebelling against the rule of God and not trusting in his provision.
For a modern Christian, it is not always easy to identify when something is stealing. For example, accessing another persons information or creative work on the internet, or cleverly structuring your finances to reduce your amount of income tax payable. The legal system is not necessarily a clear guide either as it struggles to keep pace with technological change.
Paul argues in Romans 13:9 that this command is summed up in the one rule of loving your neighbour as ourself. That is, “love is the fulfillment of the law” (Romans 13:10, and see Jesus comments in Matt 7:12). It is not enough then for us just not take from others, but we are to love others in actively pursuing their best interests (1 Thess 5:15).
So perhaps a good guide to inform how well we are keeping this command is not to ask the question “is this stealing?” but instead to ask “Am I acting in a way that seeks the best interests of the other party” – whether that be in regards to property, your investment of time, or your resources etc.
To seek the best interests of others always is also a way to guide your heart against failing to trust in God’s provision, as it takes your focus away from your own perceived needs and directs you to act in response to others needs first.