In Acts 16:1-5, we find that Timothy was the son of a Jewish mother, and a Greek father, therefore, he had not been circumcised. But Timothy was well regarded in the community and Paul wanted him to go with him as he spread word of the decision in Jerusalem on what aspects of Torah, the gentile would need to observe immediately (Acts 15:19-30) as they came to their new faith in YHWH, and left their paganism.
Sidebar: Circumcision was a central and divisive issue at the time, with the believing Jewish rabbis arguing that circumcision was required for salvation. The apostles and elders came together to consider the matter at the Jerusalem Council (Acts 15:1-6). This meeting was largely held to clarify the reasons why circumcision was not necessary for salvation. In fact, Peter gave a marvelous speech that all should read and understand. It's found in Acts 16:7-11.
Returning to Timothy and Acts 16:3: Paul knew that the believing Jews in the communities they were traveling to would not be receptive of their message, as he, Paul, was circumcised and Timothy was not. Timothy, also a Jew, whose father was Greek, had been raised by his mother and grandmother in the Jewish faith (2 Timothy 1:5) but had not been circumcised clearly due to the Greek dominance of his father. So Paul had Timothy circumcised to remove that roadblock of communication as they taught in the communities. It mattered to the Jews, but not to the Gentiles. It's as simple as that.
Now Titus was a Greek, and in a similar situation, when Paul was going into Jerusalem with Titus to teach the believing Jews (Galatians 2:2), Titus was not circumcised because it would have given the wrong impression to both Jews and Gentiles - exactly what they were trying to avoid – that Gentiles need to be circumcised in order to obtain salvation (Galatians 2:3).
Even then, there were unbelieving Jews present who, once again, tried to ensnare Paul and Titus and negate their message, but Paul and Titus held firm in their message. (Galatians 2:4-5).