In the Bible translation world, translations are often divided over whether they adhere to what is called "formal equivalence" or "dynamic equivalence." These two philosophies describe how translators match Greek or Hebrew words to the target language. Translations that ascribe to formal equivalence will ordinarily try to substitute one Greek word for one English word, while translations that prescribe to dynamic equivalence will try to determine the meaning of a Greek word in a particular passage and pick the most appropriate English word given that particular use of the word. Let's see how this plays itself out in the real world by looking at the Greek word σάρξ and how it is translated by the ESV and NIV.
Romans 7:25b (NIV): So then, I myself in my mind am a slave to God’s law, but in my sinful nature (σαρκί) a slave to the law of sin.
Romans 7:25b (ESV): So then, I myself serve the law of God with my mind, but with my flesh (σαρκί) I serve the law of sin.
The NIV and ESV translate the same Greek word in this sentence in different ways. The NIV translates σάρξ as 'sinful nature,' while the ESV translates it as 'flesh.' So which is more accurate? The answer is that it depends. The problem is that the Greek word σάρξ does not map onto any one English word perfectly. Although the dictionaries will often give the meaning as 'flesh,' the word 'flesh' in English does not have the same range of meaning as σάρξ in Greek.
In particular, in normal everyday English, we would not use the word 'flesh' to refer to a morally bad part of ourselves. Even if Christians normally speak this way, English speakers outside of Christian circles would never say that their 'flesh' made them steal something, for example. In Paul's Greek, this was an acceptable use of the word σάρξ, but since it is not ordinary English, the decided that 'sinful nature' was a more accurate way to express the idea trying to be conveyed in Greek. Now let's take a look at the same word in Romans 8:3, just a few verses later, and talk about the ESV's philosophy.
Romans 8:3 (NIV): For what the law was powerless to do because it was weakened by the flesh (σαρκός), God did by sending his own Son in the likeness of sinful flesh (σαρκός) to be a sin offering. And so he condemned sin in the flesh (σαρκί)
Romans 8:3 (ESV): For God has done what the law, weakened by the flesh (σαρκός), could not do. By sending his own Son in the likeness of sinful flesh (σαρκός) and for sin, he condemned sin in the flesh (σαρκί)
In Romans 8:3, we see that the same Greek word that was translated as 'sinful nature' by the NIV is now translated three times as 'flesh.' So what is going on? In this verse, the NIV translators see that the English word 'flesh' fits better because God's son came 'in the likeness of sinful flesh,' which makes perfect sense in English, since Jesus really did come "in the flesh," or as a person. Paul's first use of the word σάρξ is probably best understood as 'sinful nature,' since the law was made powerless and weakened not by 'flesh' (as in just being human) but by our 'sinful human nature.'
However, if the NIV translates the first occurrence as 'sinful nature' in Romans 8:3 (which fits the meaning better) rather than 'flesh,' the parallel with Jesus coming 'in the flesh' is lost, so they end up translating every occurrence as 'flesh.' In come the ESV translators who say that Romans 8:3 is obviously connected to Romans 7:25, so σάρξ should also be translated as 'flesh' in that verse to keep the same train of thought going. But the NIV people would argue that that obscures the verse, and Romans 8:3 is actually using the Greek word slightly differently.
In both cases, you have a trade-off. The NIV sacrifices the connection between Romans 7:25 and 8:3 for the sake of conveying the meaning of Romans 7:25 better, and the ESV sacrifices a clearer meaning of Romans 7:25 for the sake of keeping the connection between Romans 7:25 and Romans 8:3.
There isn't a "better" or "more accurate" translation here. The ESV's philosophy is to take every occurrence of σάρξ and try to translate it with one English word. They have chosen 'flesh' even though σάρξ does not mean 'flesh' - it just means something close to that in many contexts. What they gain by this is readers are able to see how Paul uses that word in slightly different ways in slightly different contexts. This is "formal equivalence" because the ESV translates into English based on the form of the Greek. Whenever they see the form σάρξ, they translate it with 'flesh' as much as possible.
The NIV's philosophy is to analyze the meaning of every occurrence of σάρξ separately and try to translate each occurrence with the most appropriate English word or phrase. Since words often have a variety of different interpretations, this results in having different translations for the same Greek word. The advantage is that the translations is often understood better by English speakers. The disadvantage is that connections that are clear in the original language of the Bible are sometimes obscured.
These are two different translations highlighting different parts of the overall message Paul is trying to convey. What is important is that, depending on the passage or what you are doing, you may want to see different things in your translation. Really, there is no translation that completely adopts either formal equivalence or dynamic equivalence. In each passage, the translators must decide which philosophy is best, but some translations will lean much more towards the formal equivalence side or the dynamic equivalence side.